# Course:SLU MATH 124: Math and Escher - Fall 2008 - Dr. Steve Harris

## Contents

- 1 General Information
- 2 Schedule
- 2.1 Week of August 25 - Euclidean Geometry
- 2.2 Week of September 1 - Symmetry of Rosettes
- 2.3 Week of September 8 - Basic Tessellations; Frieze Symmetries
- 2.4 Week of September 15 - Wallpaper Symmetries
- 2.5 Week of September 22 - Algebra of Symmetry Groups
- 2.6 Week of September 29 - More on Tessellations
- 2.7 Week of October 6 - Escher's Tessellations
- 2.8 Week of October 13 - Spherical Geometry
- 2.9 Week of October 20 - Spherical Isometries and Tessellations
- 2.10 Week of October 27 - Finishing Spherical Geometry, Introduction to Hyperbolic Geometry
- 2.11 Week of November 3 - Spherical Tessellations; Hyperbolic Geometry
- 2.12 Week of November 10 - Hyperbolic Tessellations, Normal and Ideal
- 2.13 Week of November 17 - Exam II
- 2.14 Week of November 24 - Multiple Geometries; Intro to Fractals
- 2.15 Week of December 1 - Fractals; Other Topological Ideas
- 2.16 Week of December 8 - Summaries, Conclusions, Questions

- 3 Alhambra Pictures
- 4 work area

## General Information

**Class Time:** 1:10 PM - 2:00 PM MWF

**Where:** Ritter 225

**Contact Information:**
• Office: Ritter Hall 208 (office hours: MWF, 3:00--late)
• Email: harrissg@slu.edu
• Phone: (314) 977-2439

**Books:**

• M.C. Escher: Visions of Symmetry by D. Schattschneider. W.H. Freeman and Company (1990)

**Prerequisite:** 3 years of high school mathematics or MT A 120 (College Algebra).

**Grading:**
probably two in-class exams; at least one project; various in-class worksheets and homework assignments; and a final exam

**Final:** Friday December 12 12:00 N – 1:50 PM

How to do well: Attendance and participation are extremely important. It is very hard to make up this material on one's own.

**Syllabus:**

This is an inquiry course, meaning that it is exploratory in nature. The course is structured to provide guidance in your own explorations into various facets of geometry and other branches of mathematics, using the art of Escher as a lodestone, a starting place, a goal, and an inspiration; but there is very little explicit lecturing. I will ask for mathematical endeavor, coherent writing, and artistic creativity. Both individual and group efforts will be engaged.

The emphasis will, indeed, be on exploration; the majority of class time will consist in your exploration by drawing—whether with paper or computer—various visualizations of the concepts at hand; group work is recommended for this, and groups can work very well together on homework as well. Class explorations and individual homework will form a substantial portion of the grade. There will also be one largish project, including an artistic creation and a substantial written report, (likely) two in-class exams, and a final exam; there will be a field trip to the Cathedral Basilica and perhaps other minor projects. I will expect reports to be carefully written with sentences and paragraphs.

There are a number of possible tracks this course can go down; we won’t have time to investigate all of them in detail, so choices will be made according to the way the class develops. Possible tracks:

- creative (primarily artistic)
- mathematical structure of tessellations and symmetries
- different geometries (flat, spherical, hyperbolic), as described by tessellations and other means
- group structures as seen in symmetries and motions
- impossible figures and the fourth dimension
- fractals

Aside from group work and informal homework, all your work should be your own; the University has strict policies on academic dishonesty. For reports, always cite your sources, and do not present anything as your own work if it should be credited to another.

Email is a good way to keep in touch informally, including at night. Stay in touch!

“Any student who feels that he/she may need academic accommodations in order to meet the requirements of this course-as outlined in the syllabus, due to presence of a disability, should contact the Office of Diversity and Affirmative Action. Please telephone the office at 314-977-8885, or visit DuBourg Hall Room 36. Confidentiality will be observed in all inquiries.”

### Grading

Letter grades are paramount in my grading scheme, not percentages. I'll always let you know what letter grade corresponds to a given numerical grade. Exams are weighted at 100 points, except for the final which is weighted at 200 points. As a rule, other items are weighted according to the maximum number of points available. But all these weightings are subject to individual adjustment to the benefit of each student (so if you better on the final or on a written project than on other inputs, that item is weighted more heavily in taking the average for the course grade; if you do worse on an exam than other inputs, it can count for less than otherwise).

Missing class is very detrimental to grades, especially in such an active-participation class as this. If you know you're going to be missing a class, let me know and we'll work out what to do about it. If you miss an in-class exploration, quiz, exam, or whatever, be sure to let me know why; if it's for a legitimate reason, we'll work out what to do about it, but I expect every reasonable effort to be made not to miss class.

## Schedule

The schedule below is a tentative schedule, based on the last time I taught this course. Please note that assignments and exams specified here are *not* authoritative until the day they are made in class; rather, this is merely a guide up until the history is written. (The farther into the future an entry here is, the less likely it is to be accurate.)

### Week of August 25 - Euclidean Geometry

Introduction to the course. We will start with exploring some of the properties of triangles and quardilaterals with an eye towards tesselations.

A good source for Escher's work online is here; try Galleries > Switzerland and Belgium, for instance.

**NOTE:** It may frequently be advantageous to open an Explortations page or Exercises page in a new window, instead of having it replace your current window (that way you can still see the instructions listed on this page, back in your original window). To do this on a Mac, for instance, hold the Control key down when clicking on the link, and select "Open Link in New Window".

Explorations:

- Do the Quadrilaterals Exploration (as much as possible in class; turn in)
- Do Tessellations, a first look Exploration (1-5 in class---need Schattschneider in class; 6-8 to work on at home, compare in class, then turn in)

Reading:

- Read the Fundamental Concepts with special attention to triangles, quadrilaterals and convexity.
- Schattschneider p. 1-19

Exercises:

- Polygon Exercises: turn in 4, 5, 7-11 (we will look at 3 in class).

### Week of September 1 - Symmetry of Rosettes

The concept of symmetry is fundamental to many of our explorations in this course. Here we will look at reflectional and rotational symmetry. We will examine some of Escher's prints, looking for how symmetry is used in his artwork.

Reading:

Explorations:

- Symmetric Figures Exploration ( introduction to the program Kali; play with in class for a while)
- Symmetry of Stars and Polygons Exploration (1-6 in class, turn in)
- Rotational and Reflectional Symmetry in Escher’s Prints (discuss in class)
- Celtic Art Exploration (discuss in class)

Exercises:

- Rosette Exercises
- turn in 3, 4, 5, 8, 9, 12, 14
- work on 15 in class

Other:

**Monday September 1 Labor Day: Official University Holiday**- Friday September 5 Last day to drop without a "W"

### Week of September 8 - Basic Tessellations; Frieze Symmetries

Reading:

- Frieze Patterns
- Schattschneider p. 19-34

Explorations:

- Tessellation Exploration: The Basics (read it all, turn in 7-9 in class)
- Identifying Border Patterns Exploration (do 1-6 in class, turn in)

Extra Credit

- Show why a rosette group can have mulitple reflection lines only if they all intersect at one point (see Diary)

Exercises:

- Frieze Exercises: turn in 1-4, 7, 8

### Week of September 15 - Wallpaper Symmetries

Reading:

Quiz:

- create a pattern, from a given motif, having a given
- rosette symmetry group
- frieze symmetry group

- may use any notes or print-outs, but not the computer

Explorations:

- Wallpaper Symmetry Exploration (do it all, turn in)
- Escher's Wallpaper Groups Exploration (do as much as we have time for, turn in)
- Symmetry Group for Border Pattern Exploration (select and turn in 3 of the questions) (leave to next week, if insufficient time)

Exercises:

- Wallpaper Exercises: turn in 1, 3, 4, 7

Optional (minor extra credit):

- identify the symmetry groups (ignoring color) in the 3 walls, door, doorway, and mosaic found in my Alhambra Pictures

### Week of September 22 - Algebra of Symmetry Groups

**Note**:

*Monday class cancelled*(in order to give you more time to do the Cathedral project)

Reading:

Special Group Project:

- visit the Cathedral Basilica and record symmetry groups, as described in Cathedral Fieldtrip Project
- The regular Mass schedule of the Cathedral includes a Friday service at noon, lasting about a 45 minutes, so plan around that time.

Explorations:

- Regular Triangle Symmetry Group Exploration (turn in 1 & 8)
- Tessellations: Why There Are Only Three Regular Tessellations (discuss in class, time permitting)

### Week of September 29 - More on Tessellations

Reading:

Explorations:

- Tessellations: Why There Are Only Three Regular Tessellations (look at 1-6; work on and turn in 7-10)
**Moved to next week:**playing around with Geometer's Sketchpad:- following along in class
- GSP Introduction Exploration (use this as a source for explaining the buttons, etc.)
- GSP Quadrilateral Tessellation Exploration (turn in a sketch of how to tessellate using a non-convex quadrilateral)
- techniques in GSP tessellation by a quadrilateral:
- marking a vector (from one vertex to of a figure to the analogous vertex of another figure) and translating the entire diagram by that vector
- marking center of rotation at a midpoint of an edge and rotating the entire diagram 180 degrees about that center

- techniques in GSP tessellation by a quadrilateral:

Exercises:

- 7-10 of the Exploration Tessellations: Why There Are Only Three Regular Tessellations.

### Week of October 6 - Escher's Tessellations

**Friday: Exam 1** (see Diary Week 6 for description)

Reading:

Explorations:

- Escher Tessellations Using Geometer’s Sketchpad (see what you can make with this; save as jpeg what you think looks good and send to mailto:harrissg@slu.edu )
**moved to next week**: Escher-Like Tessellations Explorations (turn in the sheet your team likes best)

Exercises:

- Polygonal Tessellation Exercises: turn in 3, 6-9, 15

### Week of October 13 - Spherical Geometry

Project:

- Tessellation Art Project
- preliminary sketches due by Friday, Oct. 24
- final version due by Friday, Nov. 7

Readings:

Explorations:

- Spherical Easel Exploration (turn in 1-6, think about the others)
- Spherical Geometry Exploration (turn in 1-7)
**moved to next week:**Spherical Geometry: Polygons (turn in 1-4, 6, 7)

Exercises:

**(Give these a start, but they won't be due till end of next week.**) Spherical Geometry Exercises 1, 3-8

### Week of October 20 - Spherical Isometries and Tessellations

Monday, October 20: **Fall Break, class cancelled**

Reading:

- Spherical Geometry 4, 8

Explorations:

- finish previous week's
- Spherical Geometry: Isometry Exploration (turn in 1-7)

Exercises:

- Spherical Geometry Exercises 12-16 (#16: a, b only)

### Week of October 27 - Finishing Spherical Geometry, Introduction to Hyperbolic Geometry

Reading:

Explorations:

- finish spherical explorations
**moved to next week**: Escher's Circle Limit Exploration (turn in 1-10; 11-13 to be discussed in class)

### Week of November 3 - Spherical Tessellations; Hyperbolic Geometry

Reading:

Explorations:

- Escher's Circle Limit Exploration (turn in 1-10; 11-13 to be discussed in class)
- Regular pentagonal spherical tessellations.
**below moved to next week:**- Hyperbolic Geometry Exploration (turn in 1-4)
- Hyperbolic Geometry with Noneuclid II Exploration (turn in 1-4, 7-10)

Exercises:

**moved to next week:**Hyperbolic Geometry Exercises 1-6, 9, 11, 16a.

Other: Revision of the description of the Written Report for the Tessellation Art Project:

- The Written Report should contain these elements:
- explanation of what your tessellation geometry is:
- What is the underlying regular or semi-regular polygonal tessellation?
- What changes were made to that (semi-)regular tessellation to give you your final tessellation? (What method did you employ to change the polygonal lines to depict your final motif(s)?)
- What is the symmetry group of your final tessellation (without interior details)?
- What is the symmetry group of your final design, including interior details?

- explanation of the interaction between your mathematics and your art:
- How did your choice of theme or motif(s) influence your choice of tessellation geometry or choice of alteration method?
- How did your choice of tessellation geometry or choice of alteration method influence your choice of theme or motif(s)?

- explanation of what your tessellation geometry is:
- The Written Report (revised if necessary) is not due until Monday, November 10).

### Week of November 10 - Hyperbolic Tessellations, Normal and Ideal

Reading: Hyperbolic Geometry, particularly 4.1, 5, 5.1

Explorations:

- Hyperbolic Geometry Exploration (turn in 1-4)
- Hyperbolic Geometry with Noneuclid II Exploration (turn in 1-4, 7-10)

- Hyperbolic Tessellations Exploration (turn in 2, 3, 5; 6 for extra credit)
**moved to next week**: Ideal Hyperbolic Tessellations Exploration (turn in 1-6; you may ignore the coloring, if you wish)

Exercises (due on Wednesday of next week, to accommodate Exam II on Monday):

- Hyperbolic Geometry Exercises I: 1-6, 9, 11, 16a.
- Hyperbolic Geometry Exercises II: 13a-d, 16b, 17

Special **(moved to next week)**:

**Exam II** Monday of next week:

- tessellation of the plane:
- show how to tessellate with an irregular, non-convex quadrilateral
- explain why there are only three regular tessellations

- spherical geometry:
- spherical geodesics
- for spherical triangles and other spherical n-gons:
- angle-sum
- defect
- area

- why there are only five regular tessellations

### Week of November 17 - Exam II

**Exam II**, Monday, November 17

Readings:

Explorations:

- Ideal Hyperbolic Tessellations Exploration (turn in 1-6; you may ignore the coloring, if you wish)
- Comparison between the three geometries Exploration (turn in table)
**moved to later:**Dilation Exploration (turn in 1-4; bring "Visions of Symmetry" book)**moved to later**:Iteration Exploration (turn in 1-3)

Special:

Exercise (for later):

- thoughts about the shape of the universe (1-2 page paper; to be detailed in class)

### Week of November 24 - Multiple Geometries; Intro to Fractals

Wednesday, Friday: Thanksgiving Vacation

### Week of December 1 - Fractals; Other Topological Ideas

Readings:

- Fractals
- some selection from among

Explorations:

- Dilation Exploration (turn in 1-4; bring "Visions of Symmetry" book)
- Iteration Exploration (turn in 1-3)
- Fractal Dimension Exploration (turn in 1-3)
- others: groups may choose
*ad libitum*(do at least one; anything further is extra credit):

Exercises:

- Self-Similarity Exercises 1, 2, 5, 6, 8b, 9, 12, 13, 14
- paper of at least a couple pages exploring the question,
**What is the shape of space?**- You may consider choosing among the three geometries we know, explaining what that means.
- You may consider "some of the above," "all of the above," or "none of the above," explaining what that means.
- You may consider what it would take in the way of experiments to discover the answer, or why that isn't possible.
- You may consider why this isn't even a sensible question, or what it would take to make it a sensible question.
- You may consider what the implications would be for knowing what the answer is or what it means to you personally just to contemplate the question.
**Above all else, you should think deeply and clearly express your thoughts.**- Recall from the general introduction: "Aside from group work and informal homework, all your work should be your own; the University has strict policies on academic dishonesty. For reports, always cite your sources, and do not present anything as your own work if it should be credited to another."

### Week of December 8 - Summaries, Conclusions, Questions

Class meets only on Monday.

Final Exams schedued Wednesday through next Tuesday, Dec. 10 - 16.

**Final Exam**, Friday, Dec. 12, 12:00 - 1:50

You are permitted to bring notes on how to identify frieze and wallpaper groups and one additional page of notes. The computers may not be used during the exam.

**symmetry groups**:- identification of rosette, frieze, and wallpaper groups
- multiplication of rosette or frieze transformations

**regular tessellations**:- why there are five, and only five, regular tessellations of the sphere
- need to know:
- full definition of regular tessellation
- angle-sum for spherical triangles
- angle measure in a regular spherical <math>n</math>-gon

- need to know:

- why there are five, and only five, regular tessellations of the sphere
**three geometries**:- differences
- similarities

**similarity transformations and fractals**:- describe a similarity transformation that's present in a drawing
- apply a specified similarity transformation to a figure
- What is different about fractal objects?