TOPICS FOR MT A124 Math and the Art of Escher

The topics covered will come from the following list. We typically cannot cover all of these topics.

General geometry: lines, angles, polygons

Symmetry: reflectional and rotational symmetry

Tessellations:

1. Regular tessellation (there are 3)

2. Semi-Regular Tessellation (there are 8)

3. Tetris and Pentominoes

4. Tessellating the plane with triangles

5. Tessellating the plane using quadrilaterals

Rigid Motions: Translations, Rotations and Reflections

1. Border (=Frieze) Patterns

2. Wallpaper Patterns

3. Creating Escher-like tessellation

Symmetry Groups: Rosette Symmetry Groups (Finite Groups) and Infinite groups

1. What is a group? How do we compute with one? What are subgroups?

2. Look at D2, D3 and D4 (some of the smaller dihedral groups)

      3. Look at the border pattern groups. (What type of group do we get if we only look at the translations?)

Non-Euclidean Geometry

1. What are the axioms for Euclidean Geometry?

2. Adjusting the axioms so they work on the Sphere -> Spherical Geometry

3. Rigid Motions on the Sphere: Use "Kaleidotile" program to explore geometry of the sphere. (Introduce some of the solids.)

4. Hyperbolic Geometry (See Circle Limit I, II, III, and IV for examples)

2-Dimensional versus 3-Dimensional Geometry

1. Escher's prints

2. Platonic Solids

3. Archimedean Solids

4. Prisms

5. Pyramids

6. Area and Volume

The Fourth Dimension

1. "Flatland" and "And he built a crooked house.."

2. The Hypercube or Tesseract

3. Picasso vs. Duchamp

Linear Perspective

1. Historical development of Linear Perspective (Brunelleschi, Masaccio, etc)

2. Mathematical diagrams used to aid in accurate depiction of objects on a flat plane. (Congruent Triangles and Similar Triangles)

3. Use of perspective by Escher

4. Use of perspective in Cubist Paintings

5. Impossible Figures (Penrose Triangle)

Knot Theory

1. Knot Diagrams

2. Reidemeister Moves

3. Connected Sums of Knots

4. Kirby Moves

5. Tangles

6. Border Patterns and Wallpaper patterns from knots

7. Celtic knots (Lindisfarne Gospels, Book of Kells, Book of Durrows etc.)

8. Identifying Impossible figures through the use of Knot Theory.

Topology

1. Homeomorphisms (-> Dali)

2. Surfaces : Mobius band, torus, torus knots

3. Planar Diagrams

Fractals and Chaos

1. Self-similarity: Siepinski's gadget, Koch's snowflake etc.

2. Fractals in art: Dali, Pollock, etc

3. Fractals in Architecture.