rest of term schedule :
Nov 3 - wallpaper
Nov 10 - 4x4x4 and up : 7x7x7, NxNxN, 4D, ...
Nov 17 - physics
thanksgiving
Dec 1 - presentations - last class
Nov 3
Wallpaper patterns and 2D symmetry.
readings
symmetry and group theory
plane symmetries
M.C. Escher
Wallpaper Patterns
Interactive Drawing
discussion
1. Symmetries of the 2 dimensional plane.
- Isometries;
- Euclidean Group E2.
- point group represents all rotations and refections of the plane;
- translation group represents all translations of the plane.
2. Kinds of isometry:-
- translations,
- rotations,
- reflections, and
- glide reflections.
3. Representations of symmetries of the 2D plane.
- Matrices that represent rotations and reflections.
- Vectors that represent translations.
- Glide reflections as matrices and vectors.
Recalling the group D4 represented as matrices.
- reflections that give rotations.
- Basic decoding and encoding with D4.
4. Lattices:
- Oblique
- Rectangular
- Centred rectangular
- Square
- Hexagonal.
Lattices are described more on in this link
here. Please note that oblique lattices and centred rectangular lattices are called parallelogrammatic lattices and rhombic lattices respectively.
5. Wallpaper groups are subgroups of the Euclidean group E
2 whose translation group is generated by two independent translations, and whose point group is finite. The Wikipedia page for Wallpaper groups is located
here.
6. The rotations in the point group of a wallpaper group can only have order 2, 3, 4 or 6.
7. Two wallpaper groups can be regarded as the same thing, i.e. isomorphic, if there is a correspondence between translations of the two groups, a correspondence between the rotations of the two groups, a correspondence between the refections of the two groups and a correspondence between the glide reflections between the two groups.
A picture of a wallpaper pattern from the movie Garden State can be found
here.
