The mathematics behind Escher's prints: a round trip journey from symmetry to groups and back
Summer Institute for Mathematics at the University of Washington 2007 July 9-20, 9:15-11:45 Julia Pevtsova

Course Description

Some relevant links:

• M.C. Escher Official website
• Playing with wallpaper patterns: Examples of moving tessellations
• Japanese Wallpaper patterns
• Alhambra Symmetric patterns

Lecture 1

Presentation assignments: life and work of M.C. Escher (drafts due Monday, July 16; presentations on Thursday, July 19)

• Earlier years of M.C. Escher, 1898-1941 (possible topics to include: education, Italian period/influence, "Metamorphosis")
• War and post-war period: back in Holland (possible topics to include: regular division of the plane, George Polya)
• Recognition and sucess period, 1956-1972 (possible topics to include: "Cirlce Limits" (H.S.M. Coxeter); "Waterfall", "Up and Down", (Mobius strip); "Print Gallery" ( Droste effect ) )

Some lecture notes

• Creating Wallpaper Patterns: Fries , Ghosts , more animated tessellation examples
• First problem set: 4 Rigid motions of the plane.
• Second problem set: Dihedral groups D3 and D4.
• Homework 1.

Lecture 2

Some lecture notes/handouts

• First problem set: Finish D3 and D4 problem set from Lecture 1.
• Second problem set: Dihedral groups by generators and relations. .
• Third problem set: Subgroups.

Lecture 3

Lecture outline/handouts

• Discuss Homework 1
• Finish "Subgroups of D4" problem set from Lecture 2.
• First problem set: Practice matrix multiplication.
• Homework 2.

Lecture 4

Lecture outline/handouts

• First problem set: Matrices as transformations of the plane.
• Additional problems.

Lecture 5

Lecture outline/handouts

• First problem set: Center of gravity; discrete subgroups of M.
• Tool for making patterns

Lecture 6

Lecture outline/handouts

• First problem set: Orbits of group actions.
• Second problem set : Determining point groups for the 17 wallpaper patterns.

Lecture 7

Lecture 8

Lecture outline/handouts

• Finish classification of the point groups for the 17 wallpaper groups.
• Problem set: Classification of point groups of two-dimensional crystallographic groups.
• Descriptions of the 17 wallpaper groups:
  • From wikipedia
  • From Prof. D. Joyce webpage
• Competition
• Penrose tilings (aperiodic divisions of the plane with 5-fold symmetry - time permitting)
  • Kite and Dart
  • Moving Star and Star and Sun patterns
  • And a Batman!
  • More "Penrose tilings"
• Reference: Martin Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Mathematical Games, Scientific American, January, 1977.

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