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 2013 July 29 - August 9, 9:15-11:45, SAV 132 Julia Pevtsova

Course Description

Some relevant links:

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

Lecture 1, July 29

Some Escher prints: Waterfall, also on youtube, Belvedere, Ascending descending, and a horror cartoon on the theme.

Presentation: life and work of M.C. Escher (drafts due Friday, August 2; presentations on Wednesday, August 7)

• 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 ) )

Requirements: each presentation should contain two parts: some highlights of Escher's life from the given period (mini-biography, a particular influential episode or just an amusing anecdote you can find); and going "behind" one of the prints. You can find many examples of "going behind" the Waterfall on youtube. It is preferable that the print you concentrate on is from the same chronological period as the other part of your presentation.

Presentations will be judged by an estimed "TAC committee" based on content (historical and mathematical), artistry, originality and implementation. The winning team will get Escher-related prizes on the last day of class. There will also be the "most popular" presentation selected by the students.

Assignments by teams

• Early years:
  Team Cleveland - Martijn, Jack, Lalith, and Albert
  Team NW Tree Octopudes - Frank, Lola, Udit, and Sean
  
• Middle years:
  Team L'Pel - Lucy, Peter, Eric C, Lizzie
  Team Evil League of Mugwumps - Abhi, Melanie, Erich L, and Tony
  
• Late years:
  Team Abelian Group - Andrew, Mayukha, Noah, and Michael
  Team Steamer - Adit, Alex, Peter L, and Cindy
  

Some lecture notes

• Creating Wallpaper Patterns: Fries , Ghosts , more animated tessellation examples
• Problem set 1: 4 Rigid motions of the plane.
• Homework 1.

Lecture 2, July 30

Some lecture notes/handouts

• Problem set 2 : Dihedral groups D3 and D4.
• Discussion of the D3 problem set and the homework problem from Lecture 1.
• Discussion of generators and relations in Dn, Problem set 3
• Maps and subgroups, Problem set on subgroups of D3 and D4.

Lecture 3, August 1: Maps, subgroups, generators and linear algebra

Lecture outline/handouts

• Description of the Dihedral group in terms of generators and relations, discussion of the homework set 2 from last time
• Discussion of the three homework problems: Z2xZ3 = Z6, D3 = S3, and the mysterious missing group of order 8.
• Generators of the group of isometries of the plane.
• Linear algebra: Practice matrix multiplication and identify linear transformations Problem set IV
• Homework 3.

Lecture 4, August 2: Orthogonal matrices and finite groups of rigid motions

Lecture outline/handouts

• Finish discussion of Exercise 2 in Problem set IV (geometric identifcation of linear transformations).
• Rigid motions and orthogonal matrices; Problem set V
• Finite group of rigid motions - some examples, mostly from Escher prints.
• Problem set VI on the center of gravity and the Fixed Point Theorem.
• Homework problem : Find an Escher print with the fnite group of symmetries being Dn. Which n can occur? Look at ghosts and fries - what are the finite groups of symmetry there?
• Homework 4: orbits.

Lecture 5, August 5

Lecture outline/handouts