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
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
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)
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.
- 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
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
- 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 5, August 1
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
Lecture 7: Penrose tilings
- 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
- 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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