https:///escher/api.php?action=feedcontributions&user=Barta&feedformat=atomEscherMath - User contributions [en]2020-10-27T01:40:05ZUser contributionsMediaWiki 1.34.1Course:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-12-02T18:17:51Z<p>Barta: /* Week 14 (Nov 30- Dec 4) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
<br />
<br />
==='''Week 9 (Oct 26 - Oct 30)'''===<br />
* Hyperbolic Geometry<br />
<br />
==='''Week 10 (Nov 2- Nov 6)'''===<br />
* Hyperbolic Geometry - homework (due Monday Nov 9)<br />
* '''Read [[Similarity Transformations]]'''; The '''[[Reading Assignment for Similarity Transformations]]''' is due at the beginning of class Wednesday.<br />
* Do the [[Dilation Exploration]] and [[Iteration Exploration]] (Wednesday)<br />
* '''Read [[Fractals]]''' The '''[[Reading Assignment for Fractals]]''' is due at the beginning of class Friday.<br />
* Do '''[[Self-Similarity Exploration]]''' about Escher and Dali's use of self-similarity and '''[[Escher Fractal Exploration]]''' about Iterations and fractals based on some Escher sketches.<br />
<br />
==='''Week 11 (Nov 9- Nov 13)'''===<br />
* Hyperbolic Geometry homework due Monday Nov 9<br />
* Do [[Iterative Fractals Exploration II]] about Sierpinski Triangle, Koch Snowflake, etc.(Monday)<br />
* Do [[Fractal Dimension Exploration]]: Computing the dimension of a fractal. (Wednesday)<br />
* '''Exam 2 on Spherical and Hyperbolic Geometry and Self similarity.'''<br />
<br />
==='''Week 12 (Nov 16- Nov 20)'''===<br />
* Read [[Depth and Perspective]]<br />
* Do [[Depth Exploration]] and [[Perspective Exploration]] (Monday)<br />
* Do [[Flatness Exploration]] (Wednesday)<br />
* Do '''Exam II (Friday) Postponed from last week'''<br />
<br />
==='''Week 13 (Nov 23- Nov 27)'''===<br />
* Read [[The Fourth Dimension]]<br />
* Do [[Dimensions Exploration]] (Monday)<br />
* Thanksgiving Holiday Wednesday - Friday<br />
<br />
==='''Week 14 (Nov 30- Dec 4)'''===<br />
* Read Flatland<br />
* Read [[The Mobius Band and Other Surfaces]]<br />
* Read [[History and Numbers]] <br />
<br />
* Do [[Möbius Strip Exploration]] (Monday)<br />
* Discuss Egyptian math from reading and Wiki and do [[Egyptian Numbers Exploration]] (Wednesday)<br />
<br />
==='''Week 15 (Dec 7)'''===<br />
* Read [[The Mathematical Papyri]]<br />
* Do [[Egyptian Geometry Exploration]]</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-11-18T18:51:26Z<p>Barta: /* Week 12 (Nov 16- Nov 20) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
<br />
<br />
==='''Week 9 (Oct 26 - Oct 30)'''===<br />
* Hyperbolic Geometry<br />
<br />
==='''Week 10 (Nov 2- Nov 6)'''===<br />
* Hyperbolic Geometry - homework (due Monday Nov 9)<br />
* '''Read [[Similarity Transformations]]'''; The '''[[Reading Assignment for Similarity Transformations]]''' is due at the beginning of class Wednesday.<br />
* Do the [[Dilation Exploration]] and [[Iteration Exploration]] (Wednesday)<br />
* '''Read [[Fractals]]''' The '''[[Reading Assignment for Fractals]]''' is due at the beginning of class Friday.<br />
* Do '''[[Self-Similarity Exploration]]''' about Escher and Dali's use of self-similarity and '''[[Escher Fractal Exploration]]''' about Iterations and fractals based on some Escher sketches.<br />
<br />
==='''Week 11 (Nov 9- Nov 13)'''===<br />
* Hyperbolic Geometry homework due Monday Nov 9<br />
* Do [[Iterative Fractals Exploration II]] about Sierpinski Triangle, Koch Snowflake, etc.(Monday)<br />
* Do [[Fractal Dimension Exploration]]: Computing the dimension of a fractal. (Wednesday)<br />
* '''Exam 2 on Spherical and Hyperbolic Geometry and Self similarity.'''<br />
<br />
==='''Week 12 (Nov 16- Nov 20)'''===<br />
* Read [[Depth and Perspective]]<br />
* Do [[Depth Exploration]] and [[Perspective Exploration]] (Monday)<br />
* Do [[Flatness Exploration]] (Wednesday)<br />
* Do '''Exam II (Friday) Postponed from last week'''<br />
<br />
==='''Week 13 (Nov 23- Nov 27)'''===<br />
* Read [[The Fourth Dimension]]<br />
* Do [[Dimensions Exploration]] (Monday)<br />
* Thanksgiving Holiday Wednesday - Friday<br />
<br />
==='''Week 14 (Nov 30- Dec 4)'''===<br />
* Read Flatland<br />
* Do [[Flatland Exploration]]<br />
* Read [[The Mobius Band and Other Surfaces]] and do [[Möbius Strip Exploration]] (Wednesday)<br />
* Read [[History and Numbers]] and do [[Egyptian Numbers Exploration]] (Friday)<br />
<br />
==='''Week 15 (Dec 7)'''===<br />
* Read [[The Mathematical Papyri]]<br />
* Do [[Egyptian Geometry Exploration]]</div>BartaIterative Fractals Exploration II2015-11-04T22:46:20Z<p>Barta: </p>
<hr />
<div><br />
{{Exploration}}<br />
{{Time|50}}<br />
{{Objective| <br />
A closer look at Fractals. Explore the Sierpinski triangle by creating some iterations by hand and then look at some other fractals.<br />
}}<br />
<br />
The simplest fractals are constructed by iteration. This means that we apply a certain process repeatedly. For example, start with a filled-in triangle and remove the middle fourth. Repeat this process: <br />
<br />
<br />
[[Image:Sierpinski1.png]] [[Image:Sierpinski2.png]]<br />
<br />
<br />
Here we see respectively 1 and 2 iterations of this recursive process.<br />
<br />
For every filled-in triangle, connect the midpoints of the sides and remove the middle triangle. Iterating this process produces, in the limit, the Sierpinski Gasket. The gasket is self-similar. That is, it is made up of smaller copies of itself. <br />
''"Big gaskets are made of little gaskets, The bits into which we slice 'em. And little gaskets are made of lesser gaskets And so ad infinitum."''<br />
<br />
<br />
<ol><br />
<li>Iterate the process four times, shade in the resulting triangles: <br><br />
<br />
[[Image:Sierpinski3.png|200px]] </li><br />
<br />
[[File:KochFlake.png|right|thumb|150 px| 4 iterations of the Koch snowflake]]<br />
<li> Next consider the Koch Snowflake. We can construct the Koch snowflake by the following process:<br />
* Draw an equilateral triangle<br />
* Divide each side in three equal parts<br />
* Draw an equilateral triangle on each side using the middle line segment as the base<br />
* Remove the middle line segment of the original side<br />
* Repeat!<br />
Construct your own Koch snowflake and go to at least 4 iterations.<br />
</li><br />
<br />
{{clear}}<br />
<br />
[[File:Koch-similarity-tiling.png|left|thumb|150 px| A tiling made up of two different sized Koch snowflakes]]<br />
<li> Fractals and tilings: The image of the tiling by Koch snowflakes shows a combination of two main concepts in the course.<br />
* What iteration of the Koch snowflake is being used? Are the dark and light flakes showing the same level of iteration?<br />
* What is the symmetry group of the tessellation?<br />
* Can you think of some interesting tessellations we could construct using the Siepinski triangle? Draw at least 2.<br />
</li><br />
<br />
{{clear}}<br />
<br />
<li>Next watch this 4:35 min video by Vihart named "Infinity Elephants". <br />
[https://www.youtube.com/watch?v=DK5Z709J2eo| Link to Infinity Elephants]<br />
Next draw one of the Apollonian Gaskets or one of the shapes suggested in the "Doodle Game" </li><br />
<br />
</ol><br />
{{Handin}}<br />
<br />
[[category:Similarity and Fractals Explorations]]</div>BartaIterative Fractals Exploration II2015-11-04T22:45:51Z<p>Barta: </p>
<hr />
<div><br />
{{Exploration}}<br />
{{Time|50}}<br />
{{Objective| <br />
A closer look at Fractals. Explore the Sierpinski triangle by creating some iterations by hand and then look at some other fractals.<br />
}}<br />
<br />
The simplest fractals are constructed by iteration. This means that we apply a certain process repeatedly. For example, start with a filled-in triangle and remove the middle fourth. Repeat this process: <br />
<br />
<br />
[[Image:Sierpinski1.png]] [[Image:Sierpinski2.png]]<br />
<br />
<br />
Here we see respectively 1 and 2 iterations of this recursive process.<br />
<br />
For every filled-in triangle, connect the midpoints of the sides and remove the middle triangle. Iterating this process produces, in the limit, the Sierpinski Gasket. The gasket is self-similar. That is, it is made up of smaller copies of itself. <br />
''"Big gaskets are made of little gaskets, The bits into which we slice 'em. And little gaskets are made of lesser gaskets And so ad infinitum."''<br />
<br />
<br />
<ol><br />
<li>Iterate the process four times, shade in the resulting triangles: <br><br />
<br />
[[Image:Sierpinski3.png|200px]] </li><br />
<br />
[[File:KochFlake.png|right|thumb|150 px| 4 iterations of the Koch snowflake]]<br />
<li> Next consider the Koch Snowflake. We can construct the Koch snowflake by the following process:<br />
* Draw an equilateral triangle<br />
* Divide each side in three equal parts<br />
* Draw an equilateral triangle on each side using the middle line segment as the base<br />
* Remove the middle line segment of the original side<br />
* Repeat!<br />
Construct your own Koch snowflake and go to at least 4 iterations.<br />
</li><br />
<br />
{{clear}}<br />
<br />
[[File:Koch-similarity-tiling.png|left|thumb|150 px| A tiling made up of two different sized Koch snowflakes]]<br />
<li> Fractals and tilings: The image of the tiling by Koch snowflakes shows a combination of two main concepts in the course.<br />
* What iteration of the Koch snowflake is being used? Are the dark and light flakes showing the same level of iteration?<br />
* What is the symmetry group of the tessellation?<br />
* Can you think of some interesting tessellations we could construct using the Siepinski triangle? Draw at least 2.<br />
</li><br />
<br />
{{clear}}<br />
<br />
Next watch this 4:35 min video by Vihart named "Infinity Elephants". <br />
[https://www.youtube.com/watch?v=DK5Z709J2eo| Link to Infinity Elephants]<br />
Next draw one of the Apollonian Gaskets or one of the shapes suggested in the "Doodle Game"<br />
<br />
</ol><br />
{{Handin}}<br />
<br />
[[category:Similarity and Fractals Explorations]]</div>BartaIterative Fractals Exploration II2015-11-04T22:34:07Z<p>Barta: </p>
<hr />
<div><br />
{{Exploration}}<br />
{{Time|50}}<br />
{{Objective| <br />
A closer look at Fractals. Explore the Sierpinski triangle by creating some iterations by hand and then look at some other fractals.<br />
}}<br />
<br />
The simplest fractals are constructed by iteration. This means that we apply a certain process repeatedly. For example, start with a filled-in triangle and remove the middle fourth. Repeat this process: <br />
<br />
<br />
[[Image:Sierpinski1.png]] [[Image:Sierpinski2.png]]<br />
<br />
<br />
Here we see respectively 1 and 2 iterations of this recursive process.<br />
<br />
For every filled-in triangle, connect the midpoints of the sides and remove the middle triangle. Iterating this process produces, in the limit, the Sierpinski Gasket. The gasket is self-similar. That is, it is made up of smaller copies of itself. <br />
''"Big gaskets are made of little gaskets, The bits into which we slice 'em. And little gaskets are made of lesser gaskets And so ad infinitum."''<br />
<br />
<br />
<ol><br />
<li>Iterate the process four times, shade in the resulting triangles: <br><br />
<br />
[[Image:Sierpinski3.png|200px]] </li><br />
<br />
[[File:KochFlake.png|right|thumb|150 px| 4 iterations of the Koch snowflake]]<br />
<li> Next consider the Koch Snowflake. We can construct the Koch snowflake by the following process:<br />
* Draw an equilateral triangle<br />
* Divide each side in three equal parts<br />
* Draw an equilateral triangle on each side using the middle line segment as the base<br />
* Remove the middle line segment of the original side<br />
* Repeat!<br />
Construct your own Koch snowflake and go to at least 4 iterations.<br />
</li><br />
<br />
{{clear}}<br />
<br />
[[File:Koch-similarity-tiling.png|left|thumb|150 px| A tiling made up of two different sized Koch snowflakes]]<br />
<li> Fractals and tilings: The image of the tiling by Koch snowflakes shows a combination of two main concepts in the course.<br />
* What iteration of the Koch snowflake is being used? Are the dark and light flakes showing the same level of iteration?<br />
* What is the symmetry group of the tessellation?<br />
* Can you think of some interesting tessellations we could construct using the Siepinski triangle? Draw at least 2.<br />
</li><br />
<br />
{{clear}}<br />
<br />
{| border="1"<br />
! Iteration<br />
| 0 || 1 || 2 || 3 || 4<br />
|-<br />
! Sierpinski Carpet<br />
| [[File:Menger0.PNG|85 px]] || [[File:Menger1.PNG|85 px]] || [[File:Menger2.PNG|85 px]] || [[File:Menger3.PNG|85 px]] || [[File:Menger4.PNG|85 px]]<br />
|}<br />
<br />
We construct the Sierpinski Carpet as shown above by the following procedure:<br />
* Draw a square<br />
* Divide the square in a 3 x 3 grid and remove the center square (or shade it in)<br />
* Repeat the process using the remaining 8 squares.<br />
* Repeat!<br />
<br />
</ol><br />
{{Handin}}<br />
<br />
[[category:Similarity and Fractals Explorations]]</div>BartaFile:Menger4.PNG2015-11-04T22:29:47Z<p>Barta: </p>
<hr />
<div></div>BartaFile:Menger3.PNG2015-11-04T22:28:41Z<p>Barta: </p>
<hr />
<div></div>BartaFile:Menger2.PNG2015-11-04T22:27:06Z<p>Barta: </p>
<hr />
<div></div>BartaFile:Menger1.PNG2015-11-04T22:26:17Z<p>Barta: </p>
<hr />
<div></div>BartaFile:Menger0.PNG2015-11-04T22:25:21Z<p>Barta: </p>
<hr />
<div></div>BartaIterative Fractals Exploration II2015-11-04T22:18:45Z<p>Barta: Created page with " {{Exploration}} {{Time|50}} {{Objective| A closer look at Fractals. Explore the Sierpinski triangle by creating some iterations by hand and then look at some other fractals...."</p>
<hr />
<div><br />
{{Exploration}}<br />
{{Time|50}}<br />
{{Objective| <br />
A closer look at Fractals. Explore the Sierpinski triangle by creating some iterations by hand and then look at some other fractals.<br />
}}<br />
<br />
The simplest fractals are constructed by iteration. This means that we apply a certain process repeatedly. For example, start with a filled-in triangle and remove the middle fourth. Repeat this process: <br />
<br />
<br />
[[Image:Sierpinski1.png]] [[Image:Sierpinski2.png]]<br />
<br />
<br />
Here we see respectively 1 and 2 iterations of this recursive process.<br />
<br />
For every filled-in triangle, connect the midpoints of the sides and remove the middle triangle. Iterating this process produces, in the limit, the Sierpinski Gasket. The gasket is self-similar. That is, it is made up of smaller copies of itself. <br />
''"Big gaskets are made of little gaskets, The bits into which we slice 'em. And little gaskets are made of lesser gaskets And so ad infinitum."''<br />
<br />
<br />
<ol><br />
<li>Iterate the process four times, shade in the resulting triangles: <br><br />
<br />
[[Image:Sierpinski3.png|200px]] </li><br />
<br />
[[File:KochFlake.png|right|thumb|150 px| 4 iterations of the Koch snowflake]]<br />
<li> Next consider the Koch Snowflake. We can construct the Koch snowflake by the following process:<br />
* Draw an equilateral triangle<br />
* Divide each side in three equal parts<br />
* Draw an equilateral triangle on each side using the middle line segment as the base<br />
* Remove the middle line segment of the original side<br />
* Repeat!<br />
Construct your own Koch snowflake and go to at least 4 iterations.<br />
</li><br />
<br />
{{clear}}<br />
<br />
[[File:Koch-similarity-tiling.png|left|thumb|150 px| A tiling made up of two different sized Koch snowflakes]]<br />
<li> Fractals and tilings: The image of the tiling by Koch snowflakes shows a combination of two main concepts in the course.<br />
* What iteration of the Koch snowflake is being used? Are the dark and light flakes showing the same level of iteration?<br />
* What is the symmetry group of the tessellation?<br />
* Can you think of some interesting tessellations we could construct using the Siepinski triangle? Draw at least 2.<br />
</li><br />
<br />
{{clear}}<br />
<br />
<br />
</ol><br />
{{Handin}}<br />
<br />
[[category:Similarity and Fractals Explorations]]</div>BartaFile:Koch-similarity-tiling.png2015-11-04T22:09:56Z<p>Barta: Image from Wikipedia
https://en.wikipedia.org/wiki/Koch_snowflake#/media/File:Koch_similarity_tiling.svg</p>
<hr />
<div>Image from Wikipedia<br />
https://en.wikipedia.org/wiki/Koch_snowflake#/media/File:Koch_similarity_tiling.svg</div>BartaFile:KochFlake.png2015-11-04T21:59:38Z<p>Barta: Four iterations of the Koch Snowflake. Image from Wikimedia commons.
https://en.wikipedia.org/wiki/Koch_snowflake#/media/File:KochFlake.svg</p>
<hr />
<div>Four iterations of the Koch Snowflake. Image from Wikimedia commons.<br />
<br />
https://en.wikipedia.org/wiki/Koch_snowflake#/media/File:KochFlake.svg</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-11-04T21:55:50Z<p>Barta: /* Week 11 (Nov 9- Nov 13) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
<br />
<br />
==='''Week 9 (Oct 26 - Oct 30)'''===<br />
* Hyperbolic Geometry<br />
<br />
==='''Week 10 (Nov 2- Nov 6)'''===<br />
* Hyperbolic Geometry - homework (due Monday Nov 9)<br />
* '''Read [[Similarity Transformations]]'''; The '''[[Reading Assignment for Similarity Transformations]]''' is due at the beginning of class Wednesday.<br />
* Do the [[Dilation Exploration]] and [[Iteration Exploration]] (Wednesday)<br />
* '''Read [[Fractals]]''' The '''[[Reading Assignment for Fractals]]''' is due at the beginning of class Friday.<br />
* Do '''[[Self-Similarity Exploration]]''' about Escher and Dali's use of self-similarity and '''[[Escher Fractal Exploration]]''' about Iterations and fractals based on some Escher sketches.<br />
<br />
==='''Week 11 (Nov 9- Nov 13)'''===<br />
* Hyperbolic Geometry homework due Monday Nov 9<br />
* Do [[Iterative Fractals Exploration II]] about Sierpinski Triangle, Koch Snowflake, etc.(Monday)<br />
* Do [[Fractal Dimension Exploration]]: Computing the dimension of a fractal. (Wednesday)<br />
* '''Exam 2 on Spherical and Hyperbolic Geometry and Self similarity.'''<br />
<br />
==='''Week 12 (Nov 16- Nov 20)'''===<br />
* Read [[Depth and Perspective]]<br />
* Do [[Depth Exploration]] and [[Perspective Exploration]] (Monday)<br />
* Do [[Flatness Exploration]] (Wednesday)<br />
* Do [[Impossible Exploration II]] (Friday)<br />
<br />
==='''Week 13 (Nov 23- Nov 27)'''===<br />
* Read [[The Fourth Dimension]]<br />
* Do [[Dimensions Exploration]] (Monday)<br />
* Thanksgiving Holiday Wednesday - Friday<br />
<br />
==='''Week 14 (Nov 30- Dec 4)'''===<br />
* Read Flatland<br />
* Do [[Flatland Exploration]]<br />
* Read [[The Mobius Band and Other Surfaces]] and do [[Möbius Strip Exploration]] (Wednesday)<br />
* Read [[History and Numbers]] and do [[Egyptian Numbers Exploration]] (Friday)<br />
<br />
==='''Week 15 (Dec 7)'''===<br />
* Read [[The Mathematical Papyri]]<br />
* Do [[Egyptian Geometry Exploration]]</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-11-02T13:37:05Z<p>Barta: /* Week 10 (Nov 2- Nov 6) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
<br />
<br />
==='''Week 9 (Oct 26 - Oct 30)'''===<br />
* Hyperbolic Geometry<br />
<br />
==='''Week 10 (Nov 2- Nov 6)'''===<br />
* Hyperbolic Geometry - homework (due Monday Nov 9)<br />
* '''Read [[Similarity Transformations]]'''; The '''[[Reading Assignment for Similarity Transformations]]''' is due at the beginning of class Wednesday.<br />
* Do the [[Dilation Exploration]] and [[Iteration Exploration]] (Wednesday)<br />
* '''Read [[Fractals]]''' The '''[[Reading Assignment for Fractals]]''' is due at the beginning of class Friday.<br />
* Do '''[[Self-Similarity Exploration]]''' about Escher and Dali's use of self-similarity and '''[[Escher Fractal Exploration]]''' about Iterations and fractals based on some Escher sketches.<br />
<br />
==='''Week 11 (Nov 9- Nov 13)'''===<br />
* Hyperbolic Geometry homework due Monday Nov 9<br />
* Do [[Iterative Fractals Exploration]] about Sierpinski Triangle, Koch Snowflake, etc.(Monday)<br />
* Do [[Fractal Dimension Exploration]]: Computing the dimension of a fractal. (Wednesday)<br />
* '''Exam 2 on Spherical and Hyperbolic Geometry and Self similarity.'''<br />
<br />
==='''Week 12 (Nov 16- Nov 20)'''===<br />
* Read [[Depth and Perspective]]<br />
* Do [[Depth Exploration]] and [[Perspective Exploration]] (Monday)<br />
* Do [[Flatness Exploration]] (Wednesday)<br />
* Do [[Impossible Exploration II]] (Friday)<br />
<br />
==='''Week 13 (Nov 23- Nov 27)'''===<br />
* Read [[The Fourth Dimension]]<br />
* Do [[Dimensions Exploration]] (Monday)<br />
* Thanksgiving Holiday Wednesday - Friday<br />
<br />
==='''Week 14 (Nov 30- Dec 4)'''===<br />
* Read Flatland<br />
* Do [[Flatland Exploration]]<br />
* Read [[The Mobius Band and Other Surfaces]] and do [[Möbius Strip Exploration]] (Wednesday)<br />
* Read [[History and Numbers]] and do [[Egyptian Numbers Exploration]] (Friday)<br />
<br />
==='''Week 15 (Dec 7)'''===<br />
* Read [[The Mathematical Papyri]]<br />
* Do [[Egyptian Geometry Exploration]]</div>BartaReading Assignment for Fractals2015-11-02T13:19:19Z<p>Barta: Created page with " # Explain what it means for a figure to be self similar. # What is a fractal? # Are all fractals self-similar? # Are all self similar objects fractals? # We will discuss fra..."</p>
<hr />
<div><br />
# Explain what it means for a figure to be self similar.<br />
# What is a fractal? <br />
# Are all fractals self-similar?<br />
# Are all self similar objects fractals?<br />
# We will discuss fractal dimension in more detail. What is your first impression of what a fractal dimension is?<br />
# Which example on this page did you find most interesting? Why?</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-11-02T13:14:59Z<p>Barta: /* Week 10 (Nov 2- Nov 6) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
<br />
<br />
==='''Week 9 (Oct 26 - Oct 30)'''===<br />
* Hyperbolic Geometry<br />
<br />
==='''Week 10 (Nov 2- Nov 6)'''===<br />
* Hyperbolic Geometry - homework (due Monday Nov 9)<br />
* Read [[Similarity Transformations]]; The '''[[Reading Assignment for Similarity Transformations]]''' is due at the beginning of class Wednesday.<br />
* Do the [[Dilation Exploration]] and [[Iteration Exploration]] (Wednesday)<br />
* Read [[Fractals]] The '''[[Reading Assignment for Fractals]]''' is due at the beginning of class Friday.<br />
* Do [[Self-Similarity Exploration]] about Escher and Dali's use of self-similarity and [[Escher Fractal Exploration]] about Iterations and fractals based on some Escher sketches.<br />
<br />
==='''Week 11 (Nov 9- Nov 13)'''===<br />
* Hyperbolic Geometry homework due Monday Nov 9<br />
* Do [[Iterative Fractals Exploration]] about Sierpinski Triangle, Koch Snowflake, etc.(Monday)<br />
* Do [[Fractal Dimension Exploration]]: Computing the dimension of a fractal. (Wednesday)<br />
* '''Exam 2 on Spherical and Hyperbolic Geometry and Self similarity.'''<br />
<br />
==='''Week 12 (Nov 16- Nov 20)'''===<br />
* Read [[Depth and Perspective]]<br />
* Do [[Depth Exploration]] and [[Perspective Exploration]] (Monday)<br />
* Do [[Flatness Exploration]] (Wednesday)<br />
* Do [[Impossible Exploration II]] (Friday)<br />
<br />
==='''Week 13 (Nov 23- Nov 27)'''===<br />
* Read [[The Fourth Dimension]]<br />
* Do [[Dimensions Exploration]] (Monday)<br />
* Thanksgiving Holiday Wednesday - Friday<br />
<br />
==='''Week 14 (Nov 30- Dec 4)'''===<br />
* Read Flatland<br />
* Do [[Flatland Exploration]]<br />
* Read [[The Mobius Band and Other Surfaces]] and do [[Möbius Strip Exploration]] (Wednesday)<br />
* Read [[History and Numbers]] and do [[Egyptian Numbers Exploration]] (Friday)<br />
<br />
==='''Week 15 (Dec 7)'''===<br />
* Read [[The Mathematical Papyri]]<br />
* Do [[Egyptian Geometry Exploration]]</div>BartaReading Assignment for Similarity Transformations2015-11-02T13:13:29Z<p>Barta: </p>
<hr />
<div><br />
Carefully read the section [[Similarity Transformations]].<br />
<br />
# Describe the concept of similarity in your own words.<br />
# Describe the concept of dilation in your own words.<br />
# Describe the concept of iteration in your own words.<br />
# record questions and comments you have based on the reading.<br />
# Do problem 8a in [[Self-Similarity Exercises]]</div>BartaReading Assignment for Similarity Transformations2015-11-02T13:11:17Z<p>Barta: Created page with " Carefully read the section Similarity Transformations. 1. Describe the concepts of similarity, dilation and iteration in your own words. 2. Do problem 8a in Self-Simil..."</p>
<hr />
<div><br />
Carefully read the section [[Similarity Transformations]].<br />
<br />
1. Describe the concepts of similarity, dilation and iteration in your own words.<br />
2. Do problem 8a in [[Self-Similarity Exercises]]</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-11-02T13:07:44Z<p>Barta: /* Week 10 (Nov 2- Nov 6) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
<br />
<br />
==='''Week 9 (Oct 26 - Oct 30)'''===<br />
* Hyperbolic Geometry<br />
<br />
==='''Week 10 (Nov 2- Nov 6)'''===<br />
* Hyperbolic Geometry - homework (due Monday Nov 9)<br />
* Read [[Similarity Transformations]] the [[Reading Assignment for Similarity Transformations]] is due at the beginning of class Wednesday.<br />
* Do the [[Dilation Exploration]] and [[Iteration Exploration]] (Wednesday)<br />
* Read [[Fractals]]<br />
* Do [[Self-Similarity Exploration]] about Escher and Dali's use of self-similarity and [[Escher Fractal Exploration]] about Iterations and fractals based on some Escher sketches.<br />
<br />
==='''Week 11 (Nov 9- Nov 13)'''===<br />
* Hyperbolic Geometry homework due Monday Nov 9<br />
* Do [[Iterative Fractals Exploration]] about Sierpinski Triangle, Koch Snowflake, etc.(Monday)<br />
* Do [[Fractal Dimension Exploration]]: Computing the dimension of a fractal. (Wednesday)<br />
* '''Exam 2 on Spherical and Hyperbolic Geometry and Self similarity.'''<br />
<br />
==='''Week 12 (Nov 16- Nov 20)'''===<br />
* Read [[Depth and Perspective]]<br />
* Do [[Depth Exploration]] and [[Perspective Exploration]] (Monday)<br />
* Do [[Flatness Exploration]] (Wednesday)<br />
* Do [[Impossible Exploration II]] (Friday)<br />
<br />
==='''Week 13 (Nov 23- Nov 27)'''===<br />
* Read [[The Fourth Dimension]]<br />
* Do [[Dimensions Exploration]] (Monday)<br />
* Thanksgiving Holiday Wednesday - Friday<br />
<br />
==='''Week 14 (Nov 30- Dec 4)'''===<br />
* Read Flatland<br />
* Do [[Flatland Exploration]]<br />
* Read [[The Mobius Band and Other Surfaces]] and do [[Möbius Strip Exploration]] (Wednesday)<br />
* Read [[History and Numbers]] and do [[Egyptian Numbers Exploration]] (Friday)<br />
<br />
==='''Week 15 (Dec 7)'''===<br />
* Read [[The Mathematical Papyri]]<br />
* Do [[Egyptian Geometry Exploration]]</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-11-02T13:01:45Z<p>Barta: /* Week 11 (Nov 9- Nov 13) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
<br />
<br />
==='''Week 9 (Oct 26 - Oct 30)'''===<br />
* Hyperbolic Geometry<br />
<br />
==='''Week 10 (Nov 2- Nov 6)'''===<br />
* Hyperbolic Geometry - homework (due Monday Nov 9)<br />
* Read [[Similarity Transformations]]<br />
* Do the [[Dilation Exploration]] and [[Iteration Exploration]] (Wednesday)<br />
* Read [[Fractals]]<br />
* Do [[Self-Similarity Exploration]] about Escher and Dali's use of self-similarity and [[Escher Fractal Exploration]] about Iterations and fractals based on some Escher sketches.<br />
<br />
==='''Week 11 (Nov 9- Nov 13)'''===<br />
* Hyperbolic Geometry homework due Monday Nov 9<br />
* Do [[Iterative Fractals Exploration]] about Sierpinski Triangle, Koch Snowflake, etc.(Monday)<br />
* Do [[Fractal Dimension Exploration]]: Computing the dimension of a fractal. (Wednesday)<br />
* '''Exam 2 on Spherical and Hyperbolic Geometry and Self similarity.'''<br />
<br />
==='''Week 12 (Nov 16- Nov 20)'''===<br />
* Read [[Depth and Perspective]]<br />
* Do [[Depth Exploration]] and [[Perspective Exploration]] (Monday)<br />
* Do [[Flatness Exploration]] (Wednesday)<br />
* Do [[Impossible Exploration II]] (Friday)<br />
<br />
==='''Week 13 (Nov 23- Nov 27)'''===<br />
* Read [[The Fourth Dimension]]<br />
* Do [[Dimensions Exploration]] (Monday)<br />
* Thanksgiving Holiday Wednesday - Friday<br />
<br />
==='''Week 14 (Nov 30- Dec 4)'''===<br />
* Read Flatland<br />
* Do [[Flatland Exploration]]<br />
* Read [[The Mobius Band and Other Surfaces]] and do [[Möbius Strip Exploration]] (Wednesday)<br />
* Read [[History and Numbers]] and do [[Egyptian Numbers Exploration]] (Friday)<br />
<br />
==='''Week 15 (Dec 7)'''===<br />
* Read [[The Mathematical Papyri]]<br />
* Do [[Egyptian Geometry Exploration]]</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-11-02T12:51:25Z<p>Barta: /* Schedule, Assignments, etc. */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
<br />
<br />
==='''Week 9 (Oct 26 - Oct 30)'''===<br />
* Hyperbolic Geometry<br />
<br />
==='''Week 10 (Nov 2- Nov 6)'''===<br />
* Hyperbolic Geometry - homework (due Monday Nov 9)<br />
* Read [[Similarity Transformations]]<br />
* Do the [[Dilation Exploration]] and [[Iteration Exploration]] (Wednesday)<br />
* Read [[Fractals]]<br />
* Do [[Self-Similarity Exploration]] about Escher and Dali's use of self-similarity and [[Escher Fractal Exploration]] about Iterations and fractals based on some Escher sketches.<br />
<br />
==='''Week 11 (Nov 9- Nov 13)'''===<br />
* Hyperbolic Geometry homework due Monday Nov 9<br />
* Do [[Iterative Fractals Exploration]] about Sierpinski Triangle, Koch Snowflake, etc.(Monday)<br />
* Do [[Fractal Dimension Exploration]]: Computing the dimension of a fractal. (Wednesday)<br />
* '''Exam 2 on Spherical and Hyperbolic Geometry and Self similarity.'''</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-10-16T16:17:11Z<p>Barta: /* Week 8 (Oct 12 - Oct 16) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] <br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
* Wednesday:<br />
* Friday:</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-10-09T00:24:35Z<p>Barta: /* Week 7 (Oct 5 - Oct 9) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Spherical Easel Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] and [[Duality Exploration]]<br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
* Wednesday:<br />
* Friday:</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-10-04T18:40:39Z<p>Barta: /* Week 8 (Oct 12 - Oct 16) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Regular Spherical Tessellations Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] and [[Duality Exploration]]<br />
* Wednesday: '''Exam 1 On general tessellations and topics covered in Spherical geometry.'''<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
* Wednesday:<br />
* Friday:</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-10-04T18:39:54Z<p>Barta: /* Week 7 (Oct 5 - Oct 9) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Regular Spherical Tessellations Exploration]] <br />
** '''[[Reading assignment spherical geometry - Part 2]] is due.'''<br />
** '''Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22'''<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] and [[Duality Exploration]]<br />
* Wednesday: Exam 1 On general tessellations and topics covered in Spherical geometry.<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
* Wednesday:<br />
* Friday:</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-10-04T18:39:23Z<p>Barta: /* Week 8 (Oct 12 - Oct 16) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Regular Spherical Tessellations Exploration]] <br />
** [[Reading assignment spherical geometry - Part 2]] is due.<br />
** Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] and [[Duality Exploration]]<br />
* Wednesday: Exam 1 On general tessellations and topics covered in Spherical geometry.<br />
* Friday: [[Euler Characteristic Exploration]]<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
* Wednesday:<br />
* Friday:</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-10-04T18:38:16Z<p>Barta: /* Week 7 (Oct 5 - Oct 9) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: <br />
** Short lecture on angle defects and fraction of a sphere covered by triangles. <br />
** [[Spherical Triangles Exploration]]<br />
* Wednesday: <br />
** '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: <br />
** [[Regular Spherical Tessellations Exploration]] <br />
** [[Reading assignment spherical geometry - Part 2]] is due.<br />
** Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] and [[Duality Exploration]]<br />
* Wednesday: Exam 1 On general tessellations and topics covered in Spherical geometry.<br />
* Friday:<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
* Wednesday:<br />
* Friday:</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-10-04T18:37:43Z<p>Barta: /* Week 7 (Oct 5 - Oct 9) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: Short lecture on angle defects and fraction of a sphere covered by triangles. [[Spherical Triangles Exploration]]<br />
* Wednesday: '''Tessellation project is due'''<br />
** [[Spherical Geometry Exploration]]<br />
* Friday: [[Regular Spherical Tessellations Exploration]] <br />
** [[Reading assignment spherical geometry - Part 2]] is due.<br />
** Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] and [[Duality Exploration]]<br />
* Wednesday: Exam 1 On general tessellations and topics covered in Spherical geometry.<br />
* Friday:<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
* Wednesday:<br />
* Friday:</div>BartaReading assignment spherical geometry - Part 22015-10-04T18:36:00Z<p>Barta: Created page with "Carefully read the sections 6-12 and answer the following questions: * Q1. What is the Euler characteristic? * Q2. What is the dual of a tessellation? What is the dual of a c..."</p>
<hr />
<div>Carefully read the sections 6-12 and answer the following questions:<br />
<br />
* Q1. What is the Euler characteristic?<br />
* Q2. What is the dual of a tessellation? What is the dual of a cube?<br />
* Q3. What do we know about translations and rotations on the sphere? Are they related?<br />
* Q4. Follow the link to the Wikipedia page about symmetry groups of the sphere. The page divides the symmetries in different types. What are they?<br />
* Q5. What is most surprising to you about spherical geometry after reading the spherical geometry section?</div>BartaReading assignment spherical geometry - Part 12015-10-04T18:30:09Z<p>Barta: Created page with "Carefully read the sections 1, 2, 3 and 4 and answer the following questions: * Q1. What are geodesics? Why are they important? * Q2. What are antipodal points? * Q3. What do..."</p>
<hr />
<div>Carefully read the sections 1, 2, 3 and 4 and answer the following questions:<br />
<br />
* Q1. What are geodesics? Why are they important?<br />
* Q2. What are antipodal points?<br />
* Q3. What do we know about the sum of the angles if a triangle on the sphere?<br />
* Q4. What is the defect of a triangle? What would the defect of a quadrilateral be? Why?<br />
* Q5. How many regular tessellation of the sphere do we have? How does that compare to Euclidean geometry?<br />
* Q6. What are Platonic Solids and how are they related to spherical geometry?</div>BartaSpherical Geometry2015-10-04T18:27:12Z<p>Barta: /* Explorations */</p>
<hr />
<div>[[Image:Armillary-Sphere.jpg|right|thumb|300 px|[[wikipedia:Armillary_Sphere|Armillary Sphere]] at the Beijing Ancient Observatory.]]<br />
<br />
==Explorations==<br />
Begin learning about spherical geometry with:<br />
<br />
* [[Spherical Easel Exploration]] This exploration uses Spherical Easel (a Java applet) to explore the basics of spherical geometry.<br />
* [[Spherical Geometry Exploration]] Using a ball and markers, this is a hands on exploration of spherical geometry.<br />
* [[Spherical Geometry: Polygons]] What type of polygons exist on the sphere? Use of Spherical Easel is recommended.<br />
* [[Spherical Triangles Exploration]] Explore properties of spherical triangles with Kaleidotile.<br />
* [[Regular Spherical Tessellations Exploration]] Find the regular tessellations of the sphere.<br />
* [[Spherical Geometry: Isometry Exploration]] Explore the properties of translations, rotations, relfections and glide reflections on the sphere.<br />
* [[Spherical versus Euclidean Polygons Exploration]]<br />
<br />
For platonic solids and duality turn to:<br />
* [[Platonic Solids Exploration]]<br />
* [[Duality Exploration]]<br />
* [[Euler Characteristic Exploration]]<br />
<br />
Reading Assignments<br />
* [[Reading assignment spherical geometry - Part 1]]<br />
* [[Reading assignment spherical geometry - Part 2]]<br />
<br />
==Points and Lines==<br />
<br />
[[Image:globe.png|thumb|right]]<br />
Spherical geometry is nearly as old as Euclidean geometry. In fact, the word geometry means “measurement of the Earth”, and the Earth is (more or less) a sphere. The ancient Greek geometers knew the Earth was spherical, and in c235BC [[wikipedia:Eratosthenes|Eratosthenes of Cyrene]] calculated the Earth’s circumference to within about 15%. Navigation motivated the study of spherical geometry, because even 2000 years ago the fact that the earth was curved had a noticeable effect on mapmaking. Even more importantly, the sky can be (and often was) thought of as a spherical shell enclosing the earth, with sun, moon, and stars dancing about on its surface. Navigation and timekeeping required a thorough understanding of how the heavenly bodies moved, and that required spherical geometry.<br />
<br />
In geometry there are undefined terms. There are also first principles “the truth of which it is not possible to prove”<ref>D.E. Smith, History of Mathematics, Vol. II, pg. 280</ref>, according to [[wikipedia:Aristotle|Aristotle]]. These first principles are called <br />
{{define|postulates}}. In Euclidean geometry we assume that we know what is meant by “point” and “line” – these are undefined terms. To do geometry on a sphere, we need to make sense of these terms. You can try this yourself with [[Spherical Geometry Exploration]].<br />
<br />
In spherical geometry, the “points” are points on the surface of the sphere. We are not concerned with the “inside” of the sphere. A soap bubble makes a good mental image. When thinking about the Earth, it’s helpful to realize that if you shrunk the Earth and dried off the oceans with a towel, the planet would be as smooth as a pool ball, and ones elevation off the surface would be too small to notice.<br />
<br />
Lines in spherical geometry are more subtle. Since the surface is curved, there are no straight lines on it, in the usual sense of the word straight. Because of this, we use the word {{define|geodesic}} instead of line when talking about spherical geometry:<br />
<br />
;Geodesic: A {{define|geodesic}} in non-Euclidean geometry plays the role that a straight line plays in Euclidean geometry.<br />
<br />
We expect geodesics in spherical geometry to behave like straight lines in Euclidean geometry. In particular, there are two essential features of a straight line in Euclidean geometry that we expect geodesics to have:<br />
<br />
* The shortest distance between two points is a straight line.<br />
* Walking forwards, without turning, one should follow a straight line.<br />
<br />
[[Image:Sphere-halve.png|thumb|right|A [[wikipedia:Great Circle|great circle]] divides the sphere into two equal hemispheres.]]<br />
<br />
;Great Circle:A {{define|great circle}} is a circle on a sphere which divides the sphere into two equal hemispheres.<br />
<br />
A person walking on the surface of a sphere without turning will follow a great circle. The shortest distance between two points on a sphere also lies along a great circle. Because of this:<br />
[[Image:great-circles.png|thumb|right|Four geodesics (all of which wrap around the "back" side of the sphere.]]<br />
{{boxed|Geodesics in spherical geometry are great circles.}}<br />
<br />
We will treat geodesics in spherical geometry as we treat straight lines in Euclidean geometry.<br />
<br />
Consider the statement “two points determine a line”. This is a postulate of Euclidean geometry, which means we accept its truth without proof. In spherical geometry, it is not quite true. Consider the Earth’s North and South poles. These points are joined by many great circles, which are known as [[wikipedia:Meridian (geography)|meridians]] or lines of longitude. In fact, leaving the North pole in any direction and heading straight will take you to the South pole along a geodesic. The North and South pole are not the only points with this property:<br />
<br />
;Antipodal points:Two points which are opposite each other on the sphere are called {{define|antipodal points}}.<br />
<br />
In spherical geometry, we can say “two points determine a geodesic, unless they are antipodal points, in which case there are infinitely many geodesics joining them”. This is less elegant than Euclidean geometry but fairly typical for spherical geometry, where there are often exceptions for antipodal points.<br />
<br />
;Geodesic segment: A {{define|geodesic segment}} is an arc of a geodesic and its two endpoints.<br />
<br />
When saying “two points determine a line”, one usually thinks of the line segment joining the two points. On a sphere, two points lying on a geodesic create ''two'' geodesic segments since the geodesics are circles. Unless the points are antipodal, there will be a short segment and a long segment which “goes around the back of the sphere”.<br />
<br />
==Angle Sum and Area==<br />
;Spherical Polygon: A {{define|polygon}} in spherical geometry is a sequence of points and geodesic segments joining those points. The geodesic segments are called the {{define|sides}} of the polygon. <br />
<br />
A {{define|triangle}} in spherical geometry is a polygon with three sides, a quadrilateral is a polygon with four sides, and so on, as in Euclidean geometry.<br />
<br />
One fundamental result of Euclidean geometry is that the sum of the angles in any triangle is 180°. To see this, we used properties of parallel lines. However, in spherical geometry there are no parallel lines, because any pair of geodesics intersect at two (antipodal) points. Instead, in spherical geometry we have:<br />
<br />
{{boxed|The sum of the angles in any spherical triangle is more than 180°.}}<br />
<br />
[[Image:flat-under-spherical.png|thumb|right|Spherical triangles bulge out from the corresponding flat triangle.]]<br />
To justify this statement, take a spherical triangle and then draw a flat triangle with the same vertices, as in the figure. The flat triangle has angle sum 180°, and since the spherical triangle bulges out from the flat one, its angles must be larger.<br />
<br />
;Defect: The {{define|defect}} of a spherical triangle is (angle sum of the triangle) - 180°.<br />
<br />
The more area a triangle covers, the more it bulges, the more its angles differ from a Euclidean triangle, and the larger its defect. There is a direct mathematical relationship between a triangle’s area and its defect. We measure the area as a fraction of the total area of the sphere, and find that <br />
the fraction of the sphere covered by a triangle is the triangle's defect divided by 720°. As a formula:<br />
<br />
<!-- This should use Template:Boxed, but it's not working with TeX for some reason. --><br />
<center><br />
Area fraction = <math>\frac{\text{defect}}{720^\circ}</math><br />
</center><br />
<br />
The [[Spherical Triangles Exploration]] should help you understand this formula.<br />
<br />
To find the actual area covered by a triangle, you need to know the radius <math>R</math> of the sphere and then use the fact that the total surface area of a sphere of radius <math>R</math> is <math>4\pi R^2</math>.<br />
<br />
[[Image:sphere-triangle909045.png|thumb|right|Spherical 90°-90°-45° triangle]]<br />
<br />
Example: The triangle shown in the figure has two 90° angles and one 45° angle. Its angle sum is 90°+90°+45° = 225°, and its defect is 225° – 180° = 45°. It covers 45/720 = 1/16 of the sphere. Can you see how 16 of these triangles would cover the whole sphere?<br />
<br />
We saw in [[Tessellations by Polygons#euclidean-angle-sum]] that a Euclidean polygon with <math>n</math> sides has angle sum<br />
<math>(n-2) \times 180^{\circ}</math>, by cutting the polygon into <br />
<math>n-2</math><br />
triangles.<br />
A spherical polygon with <math>n</math> sides can be cut in the same way into<br />
<math>n-2</math><br />
spherical triangles, each of which has angle sum more than 180°,<br />
and so the angle sum of a spherical <math>n</math>-gon is more than <math>(n-2)\times 180^\circ</math>.<br />
Put another way, the angle sum of a spherical polygon always exceeds the angle sum of a Euclidean polygon with the same number of sides. The amount (in degrees) of excess is called the defect of the polygon. The fraction of the sphere covered by a polygon is equal to its defect divided by 720°, just as for triangles.<br />
<br />
==Spherical Tessellations And Polyhedra==<br />
A tessellation of the sphere is a covering of the sphere by tiles, with no overlapping tiles and no gaps. We focus exclusively on tessellations by tiles which are polygons. As a first step, we look for regular tessellations. Look for them yourself with [[Regular Spherical Tessellations Exploration]].<br />
<br />
===Regular Polygons on the Sphere===<br />
Recall that a regular polygon is a polygon with all sides the same length and all angles equal. We keep the same definition in non-Euclidean geometry.<br />
<br />
In Euclidean geometry, the angle sum for a polygon with n sides is <math>(n-2)\times 180^\circ</math>, and this forces the corner angles of a regular <math>n</math>-gon to be <math>\frac{(n-2)\times 180^\circ}{n}</math>. This means there is only one shape of Euclidean regular <math>n</math>-gon.<br />
<br />
In spherical geometry there are many regular <math>n</math>-gons. There is a regular <math>n</math>-gon with any angle sum larger than <math>(n-2)\times 180^\circ</math> (up to a maximum size). So, there is a regular <math>n</math>-gon with any choice of corner angle larger than <math>\frac{(n-2)\times 180^\circ}{n}</math> (again, up to some maximum size). The maximum sizes aren’t as important, and are left for the exercises.<br />
<br />
This table summarizes the corner angles of some regular polygons on the sphere:<br />
{| border="1"<br />
!width="150"|Name<br />
!width="150"|Number of Sides<br />
!width="150"|Corner Angle<br />
|- <br />
| <div style="height:15px"></div> biangle || 2 || >0<br />
|- <br />
| <div style="height:15px"></div> triangle || 3 || >60°<br />
|- <br />
| <div style="height:15px"></div> quadrilateral || 4 || >90°<br />
|- <br />
| <div style="height:15px"></div> pentagon || 5 || >108°<br />
|- <br />
| <div style="height:15px"></div> hexagon || 6 || >120°<br />
|- <br />
| <div style="height:15px"></div> heptagon || 7 || >128.57...°<br />
|- <br />
| <div style="height:15px"></div> octagon || 8 || >135°<br />
|}<br />
<br />
To make a regular tessellation of the sphere, we need to pick one regular polygon and use it to cover the sphere. As with regular tessellations of the plane, the difficulty is to fit corner angles around a vertex, which requires the corner angle to divide evenly into 360°. This means that the possibilities for corner angles are 360/2 = 180°, 360/3 = 120°, 360/4 = 90°, 360/5 = 72°, 360/6 = 60°, and so on.<br />
<br />
===Regular Tessellations===<br />
[[File:Neolithic-spheres.jpg|center]]<br />
Compare the corner angles needed for tessellating and the corner angles of spherical polygons in the table. Most spherical polygons have corner angles too large to fit together at a vertex. Ignoring biangles and 180° corner angles for the moment, there are only five possibilites for regular spherical tessellations:<br />
<br />
* triangles with 72° angles, five meeting at a vertex<br />
* triangles with 90° angles, four meeting at a vertex<br />
* triangles with 120° angles, three meeting at a vertex<br />
* quadrilaterals with 120° angles, three meeting at a vertex<br />
* pentagons with 120° angles, three meeting at a vertex<br />
<br />
When making tessellations of the Euclidean plane, it was not so surprising that once we had six 60°-60°-60° triangles around one vertex we were able to fill out the rest of the plane. With spherical geometry, we can fit five 72°-72°-72° triangles around a vertex, but as we fill up the sphere with triangles we have to hope that they come together on the back and actually close up. Amazingly, in all five cases listed above, the polygons do cover the sphere and we get a regular tessellation. Here’s what they look like:<br />
<center><br />
{|<br />
! 120° Triangles !! 90° Triangles !! 72° Triangles<br />
|-<br />
| [[Image:sphere-tess-tetra.gif]] || [[Image:sphere-tess-octa.gif]] || [[Image:sphere-tess-icosa.gif]] <br />
|}<br />
{|<br />
! 120° Quadrilaterals !! 120° Pentagons<br />
|-<br />
| [[Image:sphere-tess-cube.gif]] ||[[Image:sphere-tess-dodeca.gif]]<br />
|}<br />
</center><br />
<br />
===Degenerate Regular Tessellations===<br />
[[Image:Sphere5.png|thumb|Tessellation by two 12-gons with 180° "corner" angles.]]<br />
Two strange “degenerate” types of regular tessellations show up in spherical geometry. The first is by polygons with corner angles equal to 180°. A 180° corner doesn’t look like a corner at all, and a regular n-gon with 180° corner angles simply looks like a hemisphere with n evenly spaced dots on its edge for the “vertices”. Two of these fit together to cover the sphere. One can argue about whether this should be a polygon at all, but we’ll see that it fits very nicely in a larger picture of regular tessellations and is worth including.<br />
<br />
[[Image:Sphere4.png|thumb|left|Tessellation by 7 biangles]]<br />
The other degenerate tessellation is a “beach ball” made with k biangles which have 360°/k corner angles. The beach ball with k = 7 is shown, and there is one for any choice of k > 1.<br />
<br />
{{clear}}<br />
<br />
==Platonic Solids==<br />
The five non-degenerate regular tessellations have been known for thousands of years, although in their alter-egos as polyhedra.<br />
<br />
;Polyhedron: A {{define|polyhedron}} is a three dimensional solid with a surface made of polygons. The polygons are known as the {{define|faces}} of the polyhedron.<br />
<br />
Spherical tessellations and polyhedra are closely related.<br />
Starting with a spherical tessellation by polygons, we can often replace the spherical (curved) polygons by flat polygons (that lie inside of the sphere) with the same vertices. The resulting solid is a polyhedron. Doing this for the regular tessellations of the sphere results in five polyhedra known as the {{define|Platonic solids}}:<br />
<br />
[[Image:Platonic5.png|center]]<br />
<br />
From left to right we see: Tetrahedron, Cube, Octahedron, Dodecahedron, Icosahedron.<br />
<br />
{{printable|Nets for making paper models of the Platonic solids: [http://www.korthalsaltes.com/pdf/cube_tetrahedron.pdf tetrahedron and cube], [http://www.korthalsaltes.com/pdf/octahedron.pdf octahedron], [http://www.korthalsaltes.com/pdf/dodecahedron.pdf dodecahedron], [http://www.korthalsaltes.com/pdf/icosahedron.pdf icosahedron].}}<br />
<br />
Plato did not discover these solids, but in the dialogue Timaeus he discusses the construction of the universe and (at some length) associates the cube, tetrahedron, octahedron and icosahedron with the elemental ideas of earth, fire, air and water. The dodecahedron, he claims, “God used in the delineation of the universe”.<br />
<br />
The attachment of mystical or spiritual properties to the platonic solids is, in some sense, a tribute to their mathematical perfection, and continues to this day. The 17th century astronomer Johannes Kepler wrote extensively about them, and attempted (more or less unsuccessfully) to explain the orbits of the planets as coming from the radii of a nested set of platonic solids. His reasoning was that God must have created the universe according to the platonic solids because of their mathematical perfection. In Kepler’s time, this was a somewhat heretical stance since it suggested that God was bound by rules of mathematics discovered by science.<br />
<br />
Make friends with the Platonic solids by doing the [[Platonic Solids Exploration]].<br />
<br />
==Euler Characteristic==<br />
In 1750, the Swiss mathematician Leonhard Euler discovered a remarkable formula involving the number of faces f, edges e, and vertices v of a polyhedron:<br />
<br />
{{boxed|<math>v-e + f = 2</math>}}<br />
<br />
As a first step to understanding this equation, we will calculate v, e, and f for the Platonic solids and check that <br />
<math>v-e + f = 2</math> in these cases.<br />
<br />
We can count the faces of each Platonic solid by considering the corresponding spherical tessellation. For example, in the tessellation corresponding to the dodecahedron there are three pentagons at each vertex, so that each pentagon has 120° corner angles. The five angles give an angle sum of 5*120° = 600°. Since a Euclidean pentagon has angle sum 540°, these spherical pentagons have defect equal to 600°-540° = 60°. Each pentagon therefore covers 60°/720° = 1/12 of the sphere, and so there are 12 faces on the dodecahedron.<br />
<br />
In the spherical tessellation corresponding to the octahedron, four triangles meet at a vertex. Therefore these are 90°-90°-90° triangles, which have defect 270°-180° = 90°. Each one covers 90°/720° = 1/8 of the sphere, so there are 8 faces on the octahedron.<br />
<br />
To count the edges of the dodecahedron, notice that each of the 12 faces has 5 edges. Since each edge is shared by two faces, there are 12*5/2 = 30 edges on the dodecahedron. Another way to understand this calculation is to imagine each edge cut into a left and right half. Then each face contributes 5 half-edges, and 12 * 5 *1/2 = 30.<br />
<br />
As another example, let’s count the edges of the octahedron (which you can probably do by inspection). There are eight triangles, each with three edges, so there are 8*3/2 = 12 edges in the octahedron.<br />
<br />
We’ll count the vertices in a similar manner. For the dodecahedron, each of the 12 faces has 5 vertices. Since each vertex is shared by three faces, there are 12*5/3 = 20 edges on the dodecahedron. Another way to understand this calculation is to imagine each vertex cut into three 120° wedges. Then one corner of one face is 1/3 of a vertex, and 12*5*1/3 = 20.<br />
For the octahedron, there are 8 faces with 3 vertices each, and each vertex is shared by 4 faces. There are 8*3/4 = 6 vertices.<br />
<br />
Similar calculations establish the number of faces, edges, and vertices on the tetrahedron, cube, and icosahedron. It is also possibly to simply count by inspection. We arrive at the following table:<br />
<br />
{| border="1"<br />
!width="100"|Polyhedron<br />
!width="100"|# of vertices (v)<br />
!width="100"|# of edges (e)<br />
!width="100"|# of faces (f)<br />
!width="100"|v – e + f<br />
|- <br />
| <div style="height:15px"></div> Tetrahedron || 4 || 6 || 4 || 2<br />
|- <br />
| <div style="height:15px"></div> Cube || 8 || 12 || 6 || 2<br />
|- <br />
| <div style="height:15px"></div> Octahedron || 6 || 12 || 8 || 2<br />
|- <br />
| <div style="height:15px"></div> Dodecahedron || 20 || 30 || 12 || 2<br />
|- <br />
| <div style="height:15px"></div> Icosahedron || 12 || 30 || 20 || 2<br />
|}<br />
<br />
<br />
We calculate one more example, the “tetrakis hexahedron”, which is the basis for Escher’s [[Sphere with Angels and Devils]]. Each face is a triangle, but this is not a regular tessellation of the sphere since these are not equilateral triangles. In the corresponding spherical tessellation, two of the triangle’s corners are 60° angles, since six of them together at those points. The other corner is 90° since four triangles come together at that point. The defect is 60°+60°+90° - 180° = 30°. The area fraction is 30°/720° = 1/24, so 24 of these triangles cover the sphere. Since the tetrakis hexahedron has 24 faces and each face has 3 edges, it has 24*3/2 = 36 edges.<br />
<br />
[[Image:Tetrakis.png|thumb|The Tetrakis Hexahedron and the corresponding tessellation of the sphere.]]<br />
<br />
The easiest way to count the vertices of the tetrakis hexahedron is to use <math>v - e + f = 2</math>. Since f = 24 and e = 36, v must be 14. As a check, we count another way. Each triangle contributes 1/4 of one vertex, and 1/6 each of two others. Since there are 24 triangles, the total number of vertices <math>v = 24*(1/4 + 1/6 + 1/6) = 14</math>.<br />
<br />
Having seen some evidence that v – e + f is 2, we try to make a convincing argument that it is always 2 for spherical tessellations. We write the Greek letter ''chi'' as a shorthand, <math>\chi = v - e + f</math>.<br />
<br />
The plan is to start with a sphere with one dot on it. Since there is one face (the sphere) and one vertex (the dot),<br />
<math>\chi = v - e + f = 1 - 0 + 1 = 2</math> in this case. Now we build whatever tessellation we desire by using one of the following two moves:<br />
* Move I: Add a new dot and an edge connecting it to an existing dot.<br />
* Move II: Add an edge connecting two (different) existing dots.<br />
The key point is that neither Move I or Move II changes <math>\chi</math>. Move I adds one vertex and one edge, which cancel in v – e + f. Move II adds one new edge, and cuts one face into two, creating a net increase of one face. Again, e increasing by 1 and f increasing by 1 cancel in <math>v - e + f</math>. <br />
<br />
To make this argument into a rigorous mathematical proof, we would need to argue that any spherical tessellation can be built from the one-dot-sphere via a series of Moves I and II. While not a difficult argument, it is too technical for this discussion. However, it is not hard to draw specific tessellations by hand in this way.<br />
<br />
[[Image:Torus.png|thumb|The Euler characteristic of this torus is 0.]]<br />
<br />
The quantity <math>\chi = v - e + f</math> is called the {{define|Euler characteristic}}. It is always 2 for spherical tessellations (and for polyhedra), but can actually be different from 2 on other surfaces. <span id="Torus">A {{define|torus}} is a surface with one hole</span>, for example a donut or an inner tube. A careful count of v, e, and f on a torus made of polygons shows that its Euler characteristic is 0.<br />
<br />
The number <math>\chi</math> can detect the shape of a surface without noticing size or other deformations such as stretching (e.g. a sphere stretched into a football shape). Because of this, it is considered part of a branch of mathematics called [[wikipedia:Topology|topology]].<br />
<br />
==Duality==<br />
Every tessellation by polygons has a dual. The duality process works in Euclidean geometry, non-Euclidean geometry, and even with polyhedra. We start with Euclidean geometry first, to get the idea.<br />
<br />
To find the dual to a tessellation, start with a tessellation by polygons, and put a point inside each polygon of the tessellation. Connect these new points by line segments, connecting two points when their enclosing polygons share an edge. The dual tessellation is made up of these points and the edges connecting them.<br />
<br />
[[Image:Duality.png|thumb|center|600px|From left to right: Original tessallation; Dual points; Connect neighbors; Dual tessellation.]]<br />
<br />
Sometimes the line joining two new points will have to cross through other tiles of the tessellation, but that’s fine. Another question is what to do near the edges of a tessellation, and as you can see in the example we left some polygons open to avoid this. For tessellations that cover the entire plane, and for spherical tessellations, this isn’t an issue so we won’t worry about it.<br />
<br />
In the dualizing process, we get one new vertex for every tile of the original tessellation. We also get one edge in our dual for every edge in the original, since two points are connected by an edge if their faces had an edge in common. Finally, we get one new tile in the dual for each vertex of the original. This is because the edges leaving an original vertex get turned into edges of a polygon that surrounds the original vertex.<br />
<br />
Repeating the dual returns to the original tessellation, which means that neither one is particularly more original than the other, so we just say that two tessellations are dual. For example, the tessellation by equilateral triangles and the tessellation by regular hexagons are dual:<br />
<br />
[[Image:Duality2.png|center]]<br />
<br />
Duality works for spherical tessellations and for polyhedra. With polyhedra, the process forms the dual polyhedron inside the original, but altering the sizes can lead to pleasing “compound” solids. As an example, the octahedron and cube are dual.<br />
<br />
{| style="width:700px" align="center"<br />
|- <br />
|style="width:500px" align="center"|[[Image:Dualplatonics.png|400px]]<br />
|align="center|[[Image:Temari-dual.jpg]]<br />
|-<br />
|align="center"|Octahedron inside cube; Cube-octahedron compound; Cube inside octahedron.<br />
| A Japanese temari ball showing duality of the cube (orange) and octahedron (purple). (<cite>D. Abolt</cite>)<br />
|}<br />
<br />
Practice with duality by doing the [[Duality Exploration]].<br />
<br />
[[Image:Sun_and_Moon.jpg|thumb|right|150px|M.C. Escher, <cite>Sun and Moon</cite>, 1948.]]<br />
<br />
The cube-octahedron compound is visible in the upper left corner of Escher’s [[Stars]]. The duality in Escher's [[Double Planetoid]] works on two levels. First, there is a mathematical duality between the two tetrahedra, while at the same time there is the natural duality between the urban city and the wild jungle landscape.<br />
Duality is a recurring theme in Escher's work. His early works emphasize duality using rotation or reflection symmetry. [[Scapegoat]] juxtaposes good and evil with an order 2 rotation, while [[Paradise]] depicts man and woman almost as mirror images.<br />
<br />
Many of Escher's tessellations feature two opposing figures, and are used in prints where one figure dominates half the image while the other figure dominates in the other half. For example, in [[Day and Night]], black birds flying left dominate the daytime half of the print, while white birds flying right dominate the nighttime half. In the center of the print, both sets of birds fit together and balance the print. Escher takes a similar approach in [[Sky and Water I]] and [[Sky and Water II]], though these are less developed than [[Day and Night]].<br />
<br />
As a print artist, and particularly a maker of woodcut prints, Escher developed a strong sense of the duality between figure (foreground) and ground (background). When making a woodcut, the artist carves away wood where the block will not print, leaving the printing areas untouched. In other words, to create figure, the artist removes the ground. [[Sun and Moon]] is a beautiful example of the duality between figure and ground. One can see this as a picture of grey birds flying and partially obscuring a golden sun. Alternately, it is a picture of brightly colored birds flying against the backdrop of a night sky. Switching back and forth between these interpretations takes some mental effort, but demonstrates that either set of birds functions equally well as the figure or the ground.<br />
<br />
==Symmetries in Spherical Geometry==<br />
'''Rotations are Translations:''' Spherical rotations involve spinning the sphere around an axis line that goes through the center of the sphere. A spherical rotation has two points that don’t move, where the rotation axis hits the sphere at a pair of antipodal points. For example, the Earth (idealized a bit) rotates on its axis, and the North and South poles don’t move.<br />
Translations on the sphere are exactly the same as rotations. A translation should slide along a geodesic. The geodesics are great circles, and if you slide along a great circle the sphere rotates around an axis. Picture the Earth’s equator, and as the world turns it appears that points near the equator are being translated east.<br />
<br />
[[Image:Sliderotate.png|center]]<br />
<br />
<center>''Translation along a great circle is the same as rotation around the corresponding pole.''</center><br />
<br />
Note that translations of the sphere do differ quite a bit from translations of the plane. In the Euclidean plane translations and rotations are distinct isometries, while on the sphere they can be thought of as the same rigid motion of the sphere. An added peculiarity is that on the sphere translating through a distance greater than the circumference of the sphere would result in the image circling the sphere before reaching its destination.<br />
<br />
'''Reflections:''' You can reflect a sphere using a geodesic as your reflection line. The reflection exchanges the two hemispheres. <br />
<br />
Reflections play an important role in planar geometry. It can be shown that composing reflections across parallel mirror lines results in a translation. Reflections across two intersecting lines results in a rotation about this intersection point. <br />
<br />
On the sphere we do not have any parallel lines, and hence the composition of two distinct reflections always results in a rotation about the intersection point of the two mirror lines. But by comments above it follows that we could also interpret this as a spherical translation if we wanted.<br />
<br />
<br />
'''Glide-Reflections:''' Like Euclidean geometry, the combination of a reflection and a translation is a new kind of symmetry. We saw above that translations on the sphere are really rotations, and hence a glide-reflection could also be called a rotation-reflection.<br />
<br />
==Symmetry Groups of the Sphere==<br />
[[Image:Tetrahedral-symmetry.svg|thumb|right|All of the rotations in the symmetry group of the tetrahedron.]]<br />
Symmetry groups of plane figures were completely classified into the rosette, frieze, and wallpaper groups. There are infinitely many rosette groups in two types, depending on whether reflection symmetry is present. The frieze and wallpaper groups were finite lists, harder to classify and less structured.<br />
<br />
For the sphere, the classification of symmetry groups is closely tied to the platonic solids. In fact:<br />
{{boxed|Every symmetry group of a spherical figure comes about by selecting some of the symmetries of a regular spherical tessellation.}}<br />
For example, there are two symmetry groups coming from the octahedron: One which has all symmetries of the octahedron and a second which has only the rotations, and none of the reflections. Escher's [[Sphere with Fish]] has symmetry of this reflectionless octahedron type. Two more spherical symmetry groups come from the icosahedron, in exactly the same way. Because of duality, the cube and dodecahedron contribute nothing new - they have the same symmetries as their duals. The tetrahedron contributes three: One with no reflections, one with all six possible reflections, and a third with three of the reflections.<br />
<br />
The rest of the symmetry groups possible on the sphere come from the degenerate regular tessellations. All symmetries in these groups keep one axis of the sphere fixed in place, much like the rosette groups have a fixed center point. Because there are infinitely many degenerate regular tessellations, there are also infinitely many symmetry groups derived from them. Again, the situation parallels that of rosette groups.<br />
<br />
For more details, notation, and a complete list of all these groups, see [[wikipedia:List_of_spherical_symmetry_groups|Wikipedia's list of spherical symmetry groups]].<br />
{{clear}}<br />
<br />
==Exercises==<br />
[[Spherical Geometry Exercises]]<br />
<br />
==Relevant examples from Escher's work==<br />
* [[Concentric Rinds]]<br />
* [[Sphere with Fish]], [[Sphere with Angels and Devils]], [[Sphere with Eight Grotesques]], [[Sphere with Reptiles]], [[Carved Polyhedron with Flowers]]<br />
* [[Sphere Surface with Fish]], [[Sphere Spirals]]<br />
* [[Double Planetoid]], [[Tetrahedral Planetoid]]<br />
* [[Gravitation]]<br />
* [[Four Regular Solids (Stereometric Figure)]]<br />
* [[Crystal]], [[Order and Chaos]], [[Order and Chaos II]]<br />
* [[Stars]] and [[Study for Stars]]<br />
* [[Waterfall]]<br />
* [[Flatworms]]<br />
<br />
==Related Sites==<br />
===Spherical Geometry===<br />
* [http://merganser.math.gvsu.edu/easel/ Spherical Easel] web applet for spherical geometry.<br />
* [http://math.rice.edu/~pcmi/sphere/ The Geometry of the Sphere], by John C. Polking.<br />
* [http://geometrygames.org/KaleidoTile/index.html KaleidoTile], by Jeff Weeks.<br />
* [http://www.csd.uwo.ca/~morey/CogEng/AK.html Archimedean Kaleidoscope], by Jim Morey.<br />
* [http://www.temarimath.info/studies/default.htm Temari], a Japanese textile art, by Deborah Abolt<br />
* [http://acme.com/planimeter/ Google Planimeter], measures area of triangles on the Earth.<br />
<br />
===Platonic Solids and Polyhedra===<br />
* [[wikipedia:Platonic solid]]<br />
* [http://www.georgehart.com/virtual-polyhedra/art.html Polyhedra and Art] examples through history, by George W. Hart.<br />
* [http://www.math.dartmouth.edu/%7Ematc/math5.geometry/unit6/unit6.html Platonic Solids] from [http://www.math.dartmouth.edu/%7Ematc/math5.geometry/syllabus.html Geometry in Art and Architecture] by Paul Calter.<br />
* [http://ibiblio.org/e-notes/3Dapp/Convex.htm 3D models], by Evgeny Demidov.<br />
* [http://www.korthalsaltes.com/ Paper models of polyhedra], by Korthals Altes.<br />
* [http://www.mat.puc-rio.br/~hjbortol/mathsolid/mathsolid_en.html MatHSoliD] Java applet for viewing and unfolding Platonic and Archimedean polyhedra, by E. daSilva and H. Bortolossi.<br />
* [http://www.platonicsolids.info/GIFs.htm Animated Platonic solids, with duals], from http://www.platonicsolids.info<br />
* [http://www.toonz.com/personal/todesco/java/polyhedra/theApplet.html Applet for Platonic solids with duals], by [http://www.toonz.com/personal/todesco/ G.M. Todesco].<br />
<br />
==Notes==<br />
<references /></div>BartaIntroduction to Non-Euclidean Geometry2015-10-04T18:25:12Z<p>Barta: /* Explorations and reading assignments */</p>
<hr />
<div>[[Image:Great-circle-NY-London.svg|right|thumb|300 px]]<br />
<br />
So far we have looked at what is commonly called Euclidean geometry. There are occasions where this type of geometry doesn’t get one very far. Suppose we look at a globe model of the Earth and want to measure the distance between New York and London.<br />
A ruler won't work, because the ruler will not lie flat on the sphere to measure the length. And if measuring length on the Earth is already tricky, how would you find area, even of a relatively simple shape like the state of Colorado?<br />
<br />
Geometry was initially developed to measure things: length, area and volume. Geometric measurements were used in ancient Egyptian and Babylonian times to measure the level of the Nile, to build temples, to construct the pyramids, and to measure land for taxation.<br />
The basic objects in geometry are lines, line segments, circles and angles. These four objects are the topic of the first {{define|axioms of geometry}}, sometimes called {{define|postulates}}. The fifth postulate in traditional Euclidean geometry describes parallel lines, and will prove to be the most interesting.<br />
<br />
What are the key features of lines and line segments? One important answer is that they measure the shortest distance between two points.<br />
Builders use this property to lay out straight lines at a construction site. They put two stakes in the ground and pull a piece of string taut between them.<br />
<br />
To measure the distance between two points on a sphere one might could do the same thing: put two stakes on the sphere and pull a piece of string taut between them. This procedure produces curves called {{define|geodesics}}, which are curves that minimize the distance between their endpoints. On a spherical surface such as the Earth, geodesics are segments of curves called {{define|great circles}}. On a globe, the equator and longitude lines are examples of great circles.<br />
<br />
Non-Euclidean geometry is the study of geometry on surfaces which are not flat. Because the surface is curved, there are no straight lines in the traditional sense, but these distance minimizing curves known as geodesics will play the role of straight lines in these new geometries. Then the geodesics are used as the basic object to create non-euclidean circles, triangles and other polygons.<br />
<br />
[[Image:Sphere-with-three-perp-geodesics.png|frame|right|A sphere with three geodesics.]]<br />
<br />
On the right, consider the sphere with three geodesics (solid great circles). One is the equator, and the other two cross the equator and each other at 90° angles. These great circles are faintly visible as they continue all the way around the back of the sphere, where they meet again.<br />
<br />
The three intersections of these geodesics which are visible on the front of the sphere define three geodesic segments that form a spherical triangle. Spherical triangles can behave in very strange ways. This is a 90°-90°-90° equilateral triangle - a triangle with three right angles. Such a triangle only exists on the sphere! There is no way to draw a 90°-90°-90° triangle on a piece of flat paper paper. The sum of the angles of a Euclidean triangle is always 180, and this picture shows a spherical triangle whose angles sum to 270°.<br />
<br />
Another dramatic difference between Euclidean and non-Euclidean geometry is with parallel lines. Two lines are {{define|parallel}} if they never meet, and much of high school geometry class involves playing with properties of parallel lines. However, on a sphere any two great circles will intersect in two points. This means that it is not possible to draw parallel lines on a sphere, which also eliminates all parallelograms and even squares and rectangles.<br />
<br />
In developing non-Euclidean geometry, we will rely heavily on our knowledge of Euclidean geometry for ideas, methods, and intuition. However, major differences do appear and it is important to pay special attention to the 'rules' for Euclidean geometry that these new geometries will shatter.<br />
<br />
==Explorations and reading assignments==<br />
The following reading assignment will help students critically read the section:<br />
* [[Reading Assignment Introduction to Non-Euclidean Geometry]]<br />
<br />
==Famous Early Geometers==<br />
# [[wikipedia:Pythagoras|Pythagoras]] (ca. 540 BC) Showed that in a right triangle the sum of the squares of the sides equals the square of the hypotenuse.<br />
# [[wikipedia:Plato|Plato]] (ca 380 BC) Laid the basis for formal geometry. His name is associated with the Platonic solids. Above the entrance to his school of Philosophy (the Academy) was engraved : “Let no one ignorant of geometry enter my doors”<br />
# [[wikipedia:Aristotle|Aristotle]] (ca 340 BC) The tutor of Alexander the Great, also trained many of the great geometers of the time.<br />
# [[wikipedia:Euclid|Euclid]] (ca 300 BC) The first to write down the postulates for what is now known as Euclidean geometry. He was associated with the famous School of Alexandria.<br />
# [[wikipedia:Archimedes|Archimedes]] (ca 225 BC) Pliny called him “the God of Mathematics”. He was also associated with the School of Alexandria. His name is now associated with the Archimedean solids. He was killed during the Siege of Syracuse. He was so immersed in his math that he supposedly did not notice the city being taken over by the Romans.<br />
# [[wikipedia:Eratosthenes|Eratosthenes]] (276-194 BC) Realized that the earth was round, and was able to compute a fair approximation of its circumference.<br />
# [[wikipedia:Plutarch|Plutarch]] : “God eternally geometrizes”<br />
<br />
The geometry taught in elementary, middle, and high-school is all Euclidean geometry. As you can see above, many of the results we learn about have been known for over 2000 years.<br />
Mathematicians have investigated properties of spherical triangles for almost 2000 years, but no systematic theory for geometries other than Euclidean geometry were developed until the 18th and 19th centuries. Contributors include Lambert, Saccheri, Lobachevsky, Bolyai, Gauss, and Riemann. At the end of the 19th century it was recognized that there were different systems of geometry. We will investigate spherical geometry (the geometry of the sphere) and hyperbolic geometry (as illustrated by Escher’s prints Circle Limit I-IV).</div>BartaReading Assignment Introduction to Non-Euclidean Geometry2015-10-04T18:24:49Z<p>Barta: Created page with "Write short answers to these questions: * Q1. What are axioms (also called postulates) and how many are there? * Q2. What is non-Euclidean geometry? What example is used to i..."</p>
<hr />
<div>Write short answers to these questions:<br />
<br />
* Q1. What are axioms (also called postulates) and how many are there?<br />
* Q2. What is non-Euclidean geometry? What example is used to illustrate this concept?<br />
* Q3. What is a geodesic?<br />
* Q4. What are questions you have after reading this section?<br />
* Q5/Research. Google “axioms of geometry”. What are the axioms? Write them down on a separate sheet of paper.</div>BartaIntroduction to Non-Euclidean Geometry2015-10-04T18:23:21Z<p>Barta: </p>
<hr />
<div>[[Image:Great-circle-NY-London.svg|right|thumb|300 px]]<br />
<br />
So far we have looked at what is commonly called Euclidean geometry. There are occasions where this type of geometry doesn’t get one very far. Suppose we look at a globe model of the Earth and want to measure the distance between New York and London.<br />
A ruler won't work, because the ruler will not lie flat on the sphere to measure the length. And if measuring length on the Earth is already tricky, how would you find area, even of a relatively simple shape like the state of Colorado?<br />
<br />
Geometry was initially developed to measure things: length, area and volume. Geometric measurements were used in ancient Egyptian and Babylonian times to measure the level of the Nile, to build temples, to construct the pyramids, and to measure land for taxation.<br />
The basic objects in geometry are lines, line segments, circles and angles. These four objects are the topic of the first {{define|axioms of geometry}}, sometimes called {{define|postulates}}. The fifth postulate in traditional Euclidean geometry describes parallel lines, and will prove to be the most interesting.<br />
<br />
What are the key features of lines and line segments? One important answer is that they measure the shortest distance between two points.<br />
Builders use this property to lay out straight lines at a construction site. They put two stakes in the ground and pull a piece of string taut between them.<br />
<br />
To measure the distance between two points on a sphere one might could do the same thing: put two stakes on the sphere and pull a piece of string taut between them. This procedure produces curves called {{define|geodesics}}, which are curves that minimize the distance between their endpoints. On a spherical surface such as the Earth, geodesics are segments of curves called {{define|great circles}}. On a globe, the equator and longitude lines are examples of great circles.<br />
<br />
Non-Euclidean geometry is the study of geometry on surfaces which are not flat. Because the surface is curved, there are no straight lines in the traditional sense, but these distance minimizing curves known as geodesics will play the role of straight lines in these new geometries. Then the geodesics are used as the basic object to create non-euclidean circles, triangles and other polygons.<br />
<br />
[[Image:Sphere-with-three-perp-geodesics.png|frame|right|A sphere with three geodesics.]]<br />
<br />
On the right, consider the sphere with three geodesics (solid great circles). One is the equator, and the other two cross the equator and each other at 90° angles. These great circles are faintly visible as they continue all the way around the back of the sphere, where they meet again.<br />
<br />
The three intersections of these geodesics which are visible on the front of the sphere define three geodesic segments that form a spherical triangle. Spherical triangles can behave in very strange ways. This is a 90°-90°-90° equilateral triangle - a triangle with three right angles. Such a triangle only exists on the sphere! There is no way to draw a 90°-90°-90° triangle on a piece of flat paper paper. The sum of the angles of a Euclidean triangle is always 180, and this picture shows a spherical triangle whose angles sum to 270°.<br />
<br />
Another dramatic difference between Euclidean and non-Euclidean geometry is with parallel lines. Two lines are {{define|parallel}} if they never meet, and much of high school geometry class involves playing with properties of parallel lines. However, on a sphere any two great circles will intersect in two points. This means that it is not possible to draw parallel lines on a sphere, which also eliminates all parallelograms and even squares and rectangles.<br />
<br />
In developing non-Euclidean geometry, we will rely heavily on our knowledge of Euclidean geometry for ideas, methods, and intuition. However, major differences do appear and it is important to pay special attention to the 'rules' for Euclidean geometry that these new geometries will shatter.<br />
<br />
==Explorations and reading assignments==<br />
The following reading assignment will help students critically read the section:<br />
[[Reading Assignment Introduction to Non-Euclidean Geometry]]<br />
<br />
==Famous Early Geometers==<br />
# [[wikipedia:Pythagoras|Pythagoras]] (ca. 540 BC) Showed that in a right triangle the sum of the squares of the sides equals the square of the hypotenuse.<br />
# [[wikipedia:Plato|Plato]] (ca 380 BC) Laid the basis for formal geometry. His name is associated with the Platonic solids. Above the entrance to his school of Philosophy (the Academy) was engraved : “Let no one ignorant of geometry enter my doors”<br />
# [[wikipedia:Aristotle|Aristotle]] (ca 340 BC) The tutor of Alexander the Great, also trained many of the great geometers of the time.<br />
# [[wikipedia:Euclid|Euclid]] (ca 300 BC) The first to write down the postulates for what is now known as Euclidean geometry. He was associated with the famous School of Alexandria.<br />
# [[wikipedia:Archimedes|Archimedes]] (ca 225 BC) Pliny called him “the God of Mathematics”. He was also associated with the School of Alexandria. His name is now associated with the Archimedean solids. He was killed during the Siege of Syracuse. He was so immersed in his math that he supposedly did not notice the city being taken over by the Romans.<br />
# [[wikipedia:Eratosthenes|Eratosthenes]] (276-194 BC) Realized that the earth was round, and was able to compute a fair approximation of its circumference.<br />
# [[wikipedia:Plutarch|Plutarch]] : “God eternally geometrizes”<br />
<br />
The geometry taught in elementary, middle, and high-school is all Euclidean geometry. As you can see above, many of the results we learn about have been known for over 2000 years.<br />
Mathematicians have investigated properties of spherical triangles for almost 2000 years, but no systematic theory for geometries other than Euclidean geometry were developed until the 18th and 19th centuries. Contributors include Lambert, Saccheri, Lobachevsky, Bolyai, Gauss, and Riemann. At the end of the 19th century it was recognized that there were different systems of geometry. We will investigate spherical geometry (the geometry of the sphere) and hyperbolic geometry (as illustrated by Escher’s prints Circle Limit I-IV).</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-10-04T18:20:03Z<p>Barta: /* Week 6 (Sep 28 - Oct 2) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]<br />
* Wednesday: Power outage - Class cancelled.<br />
* Friday: Explore the regular and semi-regular tessellations on the Sphere. We used Kaleidotile. <br />
<br />
==='''Week 7 (Oct 5 - Oct 9)'''===<br />
* Monday: Short lecture on angle defects and fraction of a sphere covered by triangles. [[Spherical Triangles Exploration]]<br />
* Wednesday: '''Tessellation project is due''' [[Spherical Geometry Exploration]]<br />
* Friday: [[Regular Spherical Tessellations Exploration]] <br />
** Homework Spherical geometry due: From [[Spherical Geometry Exercises]] # 3, 8, 11, 20, 22<br />
<br />
==='''Week 8 (Oct 12 - Oct 16)'''===<br />
* Monday: [[Platonic Solids Exploration]] and [[Duality Exploration]]<br />
* Wednesday: Exam 1 On general tessellations and topics covered in Spherical geometry.<br />
* Friday:<br />
<br />
==='''Week 8 (Oct 19 - Oct 23)'''===<br />
* Monday: '''Fall Break - no class'''<br />
* Wednesday:<br />
* Friday:</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-25T16:15:48Z<p>Barta: /* Week 4 (Sep 14 - 18) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]<br />
<br />
==='''Week 5 (Sep 21 - 25)'''===<br />
<br />
* Monday: [[Angles of Polygons and Regular Tessellations Exploration]]<br />
* Wednesday: Tessellations by Recognizable Figures (see handout). Based on this page: [[Escher-Like Tessellations Explorations]]<br />
* Friday: [[Islamic Patterns Exploration]]<br />
** Homework Polygonal Tessellations is due.<br />
<br />
==='''Week 6 (Sep 28 - Oct 2)'''===<br />
<br />
* Monday: Read the section on [[Tessellations by Recognizable Figures]]</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-18T16:25:58Z<p>Barta: /* Week 4 (Sep 14 - 18) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday: [[Polyominoes Exploration]]</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-18T16:24:57Z<p>Barta: /* Week 4 (Sep 14 - 18) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1<br />
** Go over Tessellations<br />
* Wednesday: Polygonal tessellations. Discuss why all parallelograms tesselate the plane, why do all triangles tessellate the plane, etc.<br />
* Friday</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-11T13:21:34Z<p>Barta: /* Week 4 (Sep 14 - 18) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** '''''Homework 2 is due''''': [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-11T13:21:12Z<p>Barta: /* Schedule, Assignments, etc. */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
** Homework 2 is assigned: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1 (due Monday)<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** Homework 2 is due: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-11T13:20:25Z<p>Barta: /* Schedule, Assignments, etc. */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Monday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will do [[Frieze Names Exploration]] problems 1 and 2 (left from last period) and start on [[Wallpaper Patterns]]<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** Homework 2 is due: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-11T13:17:53Z<p>Barta: /* Schedule, Assignments, etc. */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 7 - 11)'''===<br />
* Moday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will start on [[Wallpaper Patterns]]<br />
<br />
==='''Week 4 (Sep 14 - 18)'''===<br />
* Monday <br />
** Homework 2 is due: [[Rosette Exercises]] # 10, 12, 13 [[Frieze Exercises]] # 7, 8, 9 [[Wallpaper Exercises]] # 1</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-10T22:33:29Z<p>Barta: /* Week 3 (Sep 9 - 11) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 9 - 11)'''===<br />
* Moday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
** [[Frieze Group Exploration]]<br />
* Friday we will start on [[Wallpaper Patterns]]</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-10T22:32:02Z<p>Barta: /* Schedule, Assignments, etc. */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 9 - 11)'''===<br />
* Moday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due today are <br />
** [[Identifying Frieze Patterns Exploration]] <br />
** [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-10T22:31:03Z<p>Barta: /* Week 3 (Sep 9 - 11) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 9 - 11)'''===<br />
* Moday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.<br />
* Friday: Due are</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-09T01:15:59Z<p>Barta: /* Week 3 (Sep 9 - 11) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 9 - 11)'''===<br />
* Moday: Labor Day Holiday<br />
* Wednesday: Do [[Frieze Group Exploration]] and [[Frieze Names Exploration]] problems 1 and 2.</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-09T01:10:38Z<p>Barta: /* Week 2 (Aug 31-Sep 4) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6<br />
<br />
==='''Week 3 (Sep 9 - 11)'''===<br />
* Moday: Labor Day Holiday<br />
* Wednesday:</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-03T23:30:29Z<p>Barta: /* Week 2 (Aug 31-Sep 4) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
<br />
'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
<br />
<br />
'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
<br />
How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
<br />
<br />
'''Further Information:''' See complete Syllabus.<br />
<br />
==Schedule, Assignments, etc. ==<br />
<br />
==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
<br />
==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]<br />
* Friday<br />
** Discuss the Exploration that was due<br />
** Homework: from [[Rosette Exercises]] do # 6, 7, 8, 13, 19 and from [[Frieze Exercises]] do # 1, 3, 6</div>BartaCourse:SLU MATH 1240: Math and Escher - Fall 2015 - Dr. Anneke Bart2015-09-03T23:25:21Z<p>Barta: /* Week 2 (Aug 31-Sep 4) */</p>
<hr />
<div>==General Information==<br />
<br />
'''Class Time:''' 1:10 - 2:00 pm Monday, Wednesday, Friday<br />
<br />
'''Where:''' TBD<br />
<br />
'''Contact Information:'''<br />
• '''Office:''' Ritter Hall 227<br />
• '''Email:''' barta@slu.edu<br />
• '''Phone:''' (314) 977-2852 (I prefer email!)<br />
<br />
'''Books:'''<br />
<br />
* M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)<br />
<br />
<br />
'''Prerequisite:''' A Math Index greater than or equal to 750 or a course at the level of College Algebra.<br />
<br />
<br />
'''Course Goals'''<br />
<ol><br />
<li> Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).<br />
<li> Develop a thorough understanding of the concepts and techniques of geometry.<br />
<li> Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.<br />
<li> (Further) develop skills for working effectively with others on mathematics problems.<br />
</ol><br />
<br />
<br />
'''Grading:'''<br />
* Two exam – 10% each<br />
* Projects Portfolio – 20%<br />
** Basilica Cathedral Fieldtrip <br />
** 3 small creative projects based on the material for border patterns, wallpaper patterns, and non-Euclidean geometry. <br />
** Saint Louis Museum Fieldtrip <br />
* Homework and in-class work – 30% (Attendance is considered part of your in-class grade.)<br />
* Final – 30%<br />
<br />
'''Final:''' <br />
* The schedule for all your final exams can be found online: [http://www.slu.edu/office-of-the-university-registrar-home/final-exam-schedule final exam schedule]<br />
* For this course the Final is on '''Friday December 11 from 12 pm until 1:50 pm'''.<br />
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'''Grades:''' <br />
Grades are determined by your mastery of the material. The following cut-offs are the standard grades for mathematics courses.<br />
93-100 A, 89-92 A-, 86-88 B+, 82-85 B, 80-81 B- 77-79 C+, 70-76 C, 60-69 D, 0-59 F<br />
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'''Curve:''' <br />
I do not technically grade on a curve, but your work will of course be compared to that of your classmates, and even to students who have taken the class before you. <br />
To give an example: when evaluating answers that require an explanation, I will collect all the answers I consider “A-level” and then rank them. If the question is worth 20 points, an A is somewhere between 18 and 20 points. The best answers will receive 20 points, the next best group will receive 19 points, and the others 18. They are all awarded an A, but the best answers receive a few more points.<br />
If someone writes answers that are truly excellent, then I will award extra credit. <br />
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How to do well:<br />
Attendance and participation is extremely important. Missing class regularly causes students quite a bit of trouble. It is very hard to make up this material on ones own. <br />
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'''Further Information:''' See complete Syllabus.<br />
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==Schedule, Assignments, etc. ==<br />
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==='''Week 1 (Aug 24-28)'''===<br />
* Go over syllabus, website and BBLearn<br />
* Read [[Introduction to Symmetry]]<br />
* Class: [[Symmetry of Stars and Polygons Exploration]] (Wednesday)<br />
* Class: [[Rotational and Reflectional Symmetry in Escher’s Sketches]]<br />
* '''Homework due Monday''': [[Rosette Exercises]] # 2, 3, 4, 5, 6. Write up your answers carefully. Explain your thinking. Usually just "yes", "no" or other short answers are not adequate. Homework should be stapled if there are multiple pages.<br />
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==='''Week 2 (Aug 31-Sep 4)'''===<br />
* Monday:<br />
** [[Symmetry and Celtic Knots Exploration]] Understand what Cn and Dn groups are.<br />
** [[Symmetric Figures Exploration]] Use Silk - bottom of page - to explore the different patterns. Make sure to explore the controls.<br />
* Wednesday<br />
** We discussed the symmetry groups for Frieze patterns and how to find them.<br />
** We started on [[Identifying Frieze Patterns Exploration]]</div>Barta