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
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 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
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
- 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
Initial Escher presentation assignments are due.
- 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
- Homework 4: orbits.
Lecture 5, August 5
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
- All discrete subgroups of O are finite.
- Definition of Crystallographic groups.
- Problem set: Crystallographic restriction.
- Descriptions of the 17 wallpaper groups:
- Tool for making patterns
- Second problem set : Determining point groups for the 17 wallpaper patterns.
Wednesday, August 7: Presentations about Escher's life and work
- General content quality
- Math content
- Presentation (artistic qualities, ideas and implementation)
Thursday, Lecture 8: More on Crystallographic groups and Penrose tilings
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