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 2011 July 25 - August 5, 9:15-11:45, SMI 407, 404 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 25

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

Presentation assignments: life and work of M.C. Escher (drafts due Friday, July 29; presentations on Tuesday, August 2)

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.

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, July 26

Some lecture notes/handouts

Discussion of D3 problem set from Lecture 1 and the first homework problem.
Discussion of generators and relations in Dn, Third problem set
Maps and subgroups, Problem set on subgroups of D3 and D4.
Generators of the group of isometries of the plane

Lecture 3, July 28: Math Auction and linear algebra

Lecture outline/handouts

Linear algebra: Practice matrix multiplication and identify linear transformations Problem set 5
Math Auction: groups of order 3, 4, 5, ... for sale.
Lattice subgroup for D4.
Homework 2.

Lecture 4, July 29: Finite groups of rigid motions

Initial Escher presentation assignments are due.

Lecture outline/handouts

Rigid motions and orthogonal matrices
Finite group of rigid motions - some examples
Problem set on the center of gravity and the Fixed Point Theorem.
Classification of finite groups of rigid motions.
Discrete groups or rigid motions

Lecture 5, August 1

Lecture outline/handouts

Problem set on orbits and the proof of the Fixed Point Theorem

Discrete groups of motions; translations subgroups and point groups. Determination of point groups for groups of symmetries of some Escher prints.

Tuesday, August 2: Presentations about Escher's life and work

General content quality
Math content
Presentation/stage work
Group work

Pike Mercuoco (Chris, Kevin, Lizzie, Yejin). Theme: Early period (1898-1941);
Benjamin Franklin (Alec, Chinmay, Chrrstina, Siddharth). Theme: Early years, "Still life and street".
Reptilians (Erdong, William, William). Theme: Middle time period (1941-1956), "Relativity"
Symphalan (Alan, Harry, Sasha, Tiffany, ) Theme: Mid career (1942-1955), Regular divisions of the plane
L=DA^2 (Lev, Danning, Alex, Anvita ). Theme: later life; "Metamorphosis III"
The Cows (Jade, Kristey, Robert, Vladimir-William). Theme: The later years; "Waterfall", "Snakes", "Path of Life III".

Lecture 6: 17 Crystallographic groups

Problem set: Crystallographic restriction.
Descriptions of the 17 wallpaper groups:
- From wikipedia
- From Prof. D. Joyce webpage
- and another description, including the orbifold notation
- 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: Penrose tilings

Lecture outline

Discussion of the names of the 17 Crystallographic groups.
Wallpaper group recognition competition. Rules: correct answer wins 50 TAC bucks times the maximal order of rotation of the group under consideration. Sources of prints: M.C. Escher Official website and Math and the Art of M.C.Escher (some prints here are in high reslution).
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
- Penrose tiles competition: the largest (objective) and the prettiest (subjective) each wins 200 TAC bucks. Bonuses are possible. TACs are the jury.
- Reference: Martin Gardner, Extraordinary nonperiodic tiling that enriches the theory of tiles, Mathematical Games, Scientific American, January, 1977.

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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 2011 July 25 - August 5, 9:15-11:45, SMI 407, 404 Julia Pevtsova

Course Description

Some relevant links:

Lecture 1, July 25

Presentation assignments: life and work of M.C. Escher (drafts due Friday, July 29; presentations on Tuesday, August 2)

Some lecture notes

Lecture 2, July 26

Some lecture notes/handouts

Lecture 3, July 28: Math Auction and linear algebra

Lecture outline/handouts

Lecture 4, July 29: Finite groups of rigid motions

Lecture outline/handouts

Lecture 5, August 1

Lecture outline/handouts

Tuesday, August 2: Presentations about Escher's life and work

Lecture 6: 17 Crystallographic groups

Lecture 7: Penrose tilings

Lecture outline

Summer Institute for Mathematics at the University of Washington 2011
July 25 - August 5, 9:15-11:45, SMI 407, 404
Julia Pevtsova