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

Algebra is nothing but written geometry; L'algèbre n'est qu'une géométrie écrite;
Geometry is nothing but pictured algebra. la géométrie n'est qu'une algèbre figurée.
  Sophie Germain

Course Description

Some relevant links:

Informal lecture notes.

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)

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

Some lecture notes

Lecture 2, July 30

Some lecture notes/handouts

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

Lecture outline/handouts

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

Initial Escher presentation assignments are due.

Lecture outline/handouts

Lecture 5, August 5

Lecture outline/handouts

  • Discussion of the homework (Problem set 7) including the "Fixed Point Theorem".
  • Classification of finite groups of rigid motions.
  • Discrete groups of motions; translations subgroups and point groups.
  • Determination of point groups for groups of symmetries of some Escher prints.

    Lecture 6, August 6: 17 Crystallographic groups

    Wednesday, August 7: Presentations about Escher's life and work

    Thursday, Lecture 8: More on Crystallographic groups and Penrose tilings

    Lecture outline

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