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 2009 June 29 - July 10, 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, June 29

Some Escher prints: Waterfall, also on youtube, Belvedere, Ascending descending.

Presentation assignments: life and work of M.C. Escher (drafts due Monday, July 6; presentations on Thursday, July 9)

• 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 success 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 group D3.
• Homework 1.

Lecture 2, June 30: DIHEDRAL GROUPS

Some lecture notes/handouts

• Finish D3 problem set from Lecture 1.
• Discussion of generators and relations in Dn; students do an example of simplifying a lond element in Dn using relations
• Maps and subgroups, Problem set.

Lecture 3, July 1: LINEAR ALGEBRA

Lecture outline/handouts

• Discuss Homework 1. Math Auction: groups of order 3, 4, 5, 6 for sale.
• Finish "Normal subgroups of D3 and D4" problem set from Lecture 2.
• Linear algebra: Practice matrix multiplication.
• Homework 2.

Lecture 4, July 2: LINEAR ALGEBRA

Lecture outline/handouts

• Math Auction (revised rules): groups of order 8 for sale.
• Linear transformations of the plane Exercise set.

Lecture 5, July 6: Finite groups of rigid motions

Lecture outline/handouts

• Rigid motions as matrices Exercise set.
• Additional problems.
• Center of gravity: Finite subgroups of M.

Lecture 6: 17 Crystallographic 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
  • 17 groups as beautiful bird patterns (from Dale Seymour Publications )
• Tool for making patterns
• Second problem set : Determining point groups for the 17 wallpaper patterns.
• Competition

Lecture 7: Presentations about Escher's life and work

Criteria:
• General content
• Clarity
• Artistic quality
• Engagement of the audience
• Group work
• Math content

Presentations:
• Asians + 1. Theme: Early years and influences; "Still Life and Street", "Day and night", "Metamorphosis I".
• Neon Purple H1N1. Early Years; "Sky and Water", Tessellations.
• Enterprise. Theme: War and Post-War; "Gravitation", "Stars", "Double Planetoid"
• Tomorrow. Theme: Mid career, mathematically inspired works; "Metamorphosis series", "Relativity".
• Domination. Theme: later years; Circle limit series
• Anti-Slytherin. Theme: Later years and death. "Ascending & Descending".

Lecture 8

Lecture outline

• Penrose tilings (aperiodic divisions of the plane with 5-fold symmetry)
  • Wang Tiles
  • Kite and Dart
  • Moving Star and Star and Sun patterns
  • And a Batman!
  • More "Penrose tilings"
  • And a movie
  • Deflation
• Reference: Martin Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Mathematical Games, Scientific American, January, 1977.