# Course:SLU MATH 124: Math and Escher - Fall 2007 - Dr. Anneke Bart

## Contents

- 1 General Information
- 2 Homework and Reading Assignments
- 3 Schedule
- 3.1 Week of August 27 - Euclidean Geometry
- 3.2 Week of September 3 - Symmetry
- 3.3 Week of September 10 - Introduction to Border Patterns and Wallpaper Groups
- 3.4 Week of September 17 - Basic Tessellations
- 3.5 Week of September 24 - More Tessellations
- 3.6 Week of October 1 - Escher’s tessellation
- 3.7 Week of October 8 - Tessellation Art Project
- 3.8 Week of October 15 - Non-Euclidean Geometry: Spherical Geometry
- 3.9 Week of October 22 Spherical Geometry Continued
- 3.10 Week of October 29 Hyperbolic Geometry
- 3.11 Week of November 5 Hyperbolic Geometry Continued
- 3.12 Week of November 12 Hyperbolic Geometry, Similarity and Fractals
- 3.13 Week of November 19 More Fractals
- 3.14 Week of November 26 Fractals and Flatland
- 3.15 Week of December 3 The Fourth Dimension and the Möbius Strip
- 3.16 Final Exam

## General Information

**Class Time:** 9:00 am - 9:50 am MWF

**Where:** Ritter Hall 225

**Contact Information:**
• Office: Ritter Hall 115
• Email: barta@slu.edu
• Phone: (314) 977-2852

**Books:**

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

- Flatland: A romance of many dimensions by E.A. Abbott, Dover Publ. (1992).

- Recommended: The Magic of M.C. Escher. (On sale in the bargain section of Barnes and Nobles for $20.)

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

**Course Goals**

- Develop an intuitive understanding of geometry by looking at examples and applications in art (mainly Escher’s work, but also some other modern artists).
- Develop a thorough understanding of the concepts and techniques of geometry.
- Further develop the ability to apply your knowledge of geometry to solve unfamiliar problems.
- (Further) develop skills for working effectively with others on mathematics problems.

**Grading:**

- One exam – 20%
- Tessellation Project – 10%
- Basilica Cathedral Fieldtrip - 10%
- Flatland and the fourth dimension project - 7%
- Saint Louis Museum Fieldtrip - 8%
- Homework and in-class work – 20%
- Final – 25%

**Final:** Monday December 17 8:00 am – 9:50 am

**Grades:**
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

**Curve:**
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.
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.
If someone writes answers that are truly excellent, then I will award extra credit.

How to do well: 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.

**Further Information:** See complete Syllabus.

## Homework and Reading Assignments

- Homework 4: Due Monday October 29.
- Homework 3: Math 124 F07 Bart Homework 3. Due Friday October 26.
- Read Spherical Geometry by Wednesday October 17.
- The Tessellation Art Project (including all sketches) is due on Monday October 15 (You may turn it in on Friday October 12 if that is more convenient). Your grade will be determined based on the Grading rubric - Dr. Bart -Fall 2007
- Tessellation Art Project : Copies of at least 5 preliminary sketches are due on Wednesday October 3;
- Read Visions of Symmetry pages 31 - 52 for Monday October 1.
- Do Tessellations: Why There Are Only Three Regular Tessellations for Friday September 28
- Math 124 F07 Bart Homework 2 - Frieze patterns and wallpaper patterns. Due Friday September 21, 2007.
- Read sections about Frieze Patterns and Wallpaper Patterns by Friday Sep 14.
- Math 124 F07 Bart Homework 1 - Polygons, etc. Due Wednesday September 12, 2007.
- Read section Introduction to Symmetry Read before Monday Sep. 10.
- Friday August 30 Read Schattschneider page 1-19
- Wednesday August 29: Read the Fundamental Concepts with special attention to triangles, quadrilaterals and convexity.

## Schedule

The schedule below is a tentative schedule.

### Week of August 27 - Euclidean Geometry

Introduction to the course. The first week we will do some exploration that will show you how to explore mathematics in this course. We will look at some triangles and quadrilaterals and finish the week by taking a first look at geometric tessellations.

Explorations:

- Do the Quadrilaterals Exploration (Monday)
- Do the Tessellations, a first look Exploration (Wednesday: only did the Escher related problem. Rest of hour used to explore answers to Monday's exploration)
- Do the Triangles and Quadrilaterals Exploration

Reading:

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

### Week of September 3 - Symmetry

One of the first topics we will cover is symmetry. We will discuss bilateral (reflectional) and rotational symmetry. We will examine some of Escher's prints and discover that symmetry is often present in his artwork.

Explorations:

- Symmetry of Stars and Polygons Exploration and Rotational and Reflectional Symmetry in Escher’s Prints were covered on wednesday. The first exploration was a class exercise and the second exploration was done as a group assignment.
- Lecture on the rozette symmetry groups. Do Celtic Art Exploration (class format) and Symmetric Figures Exploration (group format).

Reading:

- Read section Introduction to Symmetry

Other:

**Monday September 3 Labor Day: Official University Holiday**- Friday September 7 Last day to drop without a "W"

### Week of September 10 - Introduction to Border Patterns and Wallpaper Groups

Explorations:

- Give a short lecture about Isometries and border patterns. Talk about the classification of border patterns. Hand out the border pattern reference sheet - I don't have students memorize the notation. Do: Frieze Names Exploration (Monday)
- Do: Identifying Border Patterns Exploration. This exploration requires a bit more creativity. Should take all hour.
- Wallpaper Symmetry Exploration and Escher's Wallpaper Groups Exploration

Reading:

### Week of September 17 - Basic Tessellations

- Monday: Discussed Homework 1 and the explorations from last Friday. Discussed in particular problems 3, 6 and 7 from the homework because the concepts will be used later on in the course.
- Exploration: Tessellation Exploration: The Basics (Wednesday)
- Polyominoes Exploration

Reading:

- Read Schattschneider page 19-34

### Week of September 24 - More Tessellations

- On Monday: Discussed questions from homework 2; Specifically the problem where we had to match Polya's notation with the crystallographic notation for wallpaper groups. The point was also made that when drawing tessellations (on an exam for instance) a pattern needs to be established.
- Wednesday: Do Checking Tessellations Exploration and do Wallpaper Exploration; Assign Tessellations: Why There Are Only Three Regular Tessellations as homework (Due Monday).
- Discussed some problems from second homework. Spent time looking at how to find wallpaper groups.

Reading:

- Read Schattschneider page 34-52
- Read Tessellations by Polygons

### Week of October 1 - Escher’s tessellation

Explorations:

- Monday: Fieldtrip to St. Louis Cathedral Basilica
- Do Escher-Like Tessellations Explorations

Reading:

### Week of October 8 - Tessellation Art Project

- Monday: Come up with ideas for the Tessellation Art Project. Have students start on the set of preliminary sketches that are part of the assignment. We will use the Sketches for the Art Project Exploration
- Wednesday: Answer any remaining questions about the Tessellation Art Project. Introduce Non-Euclidean Geometry. Discuss Axioms.
- Friday: Spherical Easel Exploration

### Week of October 15 - Non-Euclidean Geometry: Spherical Geometry

- Monday: Spherical Geometry Exploration
- Wednesday: Spherical Geometry: Polygons
- Friday: Look at Spherical Geometry Homework. The exercises for Spherical Geometry will be divided over two assignments. The first assignment covers the basics of Spherical Geometry: Math 124 F07 Bart Homework 3

Read: Spherical Geometry

### Week of October 22 Spherical Geometry Continued

- Monday October 22: Fall Break.
- Wednesday: Spherical Geometry: Isometry Exploration
- Friday: Homework #3 is due; Start on Math 124 F07 Bart Homework 4

### Week of October 29 Hyperbolic Geometry

- Monday: Introduction to Hyperbolic Geometry. Discuss the axioms.
- Wednesday: Do Escher's Circle Limit Exploration
- Friday: Hyperbolic Geometry Exploration

Fri November 2 Last Day to Withdraw

### Week of November 5 Hyperbolic Geometry Continued

- Monday:
**Exam**Carefully look at: Outline and Study Guide for Exam - Bart07 - Wednesday: Students worked another 25 minutes on Exam. Started on Hyperbolic Geometry II with NonEuclid Exploration
- Friday: Do Ideal Hyperbolic Tessellations Exploration and start on Homework 4: Hyperbolic Geometry -Bart07

### Week of November 12 Hyperbolic Geometry, Similarity and Fractals

- Monday: Hyperbolic Geometry Homework Cont'd
- Wednesday: Do Ideal Hyperbolic Tessellations Exploration
- Friday: Do Self Similarity in Advertisement Exploration in class and have students do: Self-Similarity Exploration. Make sure the self similarity exploration is done in enough detail. The completion of Print Gallery should be looked at in-depth.

### Week of November 19 More Fractals

- Monday: Self-Similarity : A first look
- Wednesday:
**Thanksgiving: Official University Holiday** - Friday:
**Thanksgiving: Official University Holiday**

### Week of November 26 Fractals and Flatland

- Monday: Fractal Dimension Exploration and read Flatland
- Wednesday: Flatland Exploration There will be a reading quiz at the beginning of class.
- Friday: Flatness Exploration

### Week of December 3 The Fourth Dimension and the Möbius Strip

- Monday: Dimensions Exploration.
- Wednesday: Möbius Strip Exploration

The Project Flatland and the Fourth Dimension and the Flatland exploration from last week are due. - Friday: Review for final. Visit the Saint Louis Art Museum.

- Monday, December 10th: No Class.

The Saint Louis Art Museum paper is due on the day of the Final - December 17, 2007.

### Final Exam

The final exam is on Monday December 17, 2007 8 a.m. - 9:50 a.m. in Ritter Hall 225 The exam is cumulative. It may help to follow this study guide: Study Guide - Final - Bart-Fall07