# Course:Harris, Fall 07: Diary Week 5

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- collected Wallpaper Exercises
- asked for Cathedral Project to be turned in this week:
- pictures (printed, published to web, burned to disk, etc.)
- symmetry group identified for each picture

- looked into algebra of symmetry groups:
- C3, consisting of
- <math>E</math> = identity/neutral element (no change)
- <math>R</math> = 120-degree rotation counterclockwise (CCW is default meaning of positive rotation)
- <math>R^2</math> = doing <math>R</math> twice
- nothing else, as repeating <math>R</math> any number of times (including backward) just results in one of the above

- D2 (symmetries of a rectange), consisting of
- <math>E</math> = identity
- <math>R</math> = 180-degree rotation
- <math>M1</math> = reflection across vertical axis
- <math>M2</math> = reflection across horizontal axis
- a multiplication table showing how to combine any of the above elements to produce one of those elements

- any row and any column in a multiplication table must contain each of the group elements

- C3, consisting of

Wed:

- use of this class in (i) my graduate course in differential geometry and (ii) my research on the boundaries of spacetime:
- want to define five types of surfaces (both for grad class and as boundaries for a certain kind of spacetime):
- this requires a symmetry group without rotations or reflections
- but we know there are only four such:
- p111 (one translation)
- p1a1 (one glide-reflection)
- p1 (two translations)
- pg (one translation, one glide-reflection)

- those produce, respectively, the cylinder, the Möbius strip, the torus, and the Klein bottle
- what about the projective plane, then? --it is formed, not from a symmetry group on the plane, but from a symmetry group on the sphere!

- class looked at the D4 symmetry group:
- the eight elements: <math>E</math>, <math>M1</math>, <math>M2</math>, <math>M3</math>, <math>M4</math>, <math>R</math>, <math>R^2</math>, <math>R^3</math>
- the easy multiplications: <math>R</math> x <math>R^2</math>, etc.
- a couple of the "interesting" multiplications: <math>M1</math> x <math>M2</math>, <math>M2</math> x <math>M1</math>
- N.B.: those two multiplications give different results!

- orientation considerations:
- the product of two mirror-reflections must be a rotation
- the product of a mirror-reflection and a rotation must be a mirror-reflection

- groups finished up the multiplication table for D4
- students should do multiplication table for D3 at home so that groups can put it together quickly on Friday
- Cathedral Project due by end of this week

Fri:

- mentioned that hints for correcting exercises about wallpaper groups are based on this method of organizing
- class finished up D4 & D3 Symmetry Group Exploration
- we looked at the symmetries of the MM frieze group:
- <math>E</math>
- <math>H</math> (horizantal reflection)
- <math>V_0</math>, <math>V_1</math>, <math>V_2</math>, etc. (many vertical reflections)
- <math>R_0</math>, <math>R_1</math>, <math>R_2</math>, etc. (many 180-degree rotations)
- <math>T</math>, <math>T^2</math>, <math>T^3</math>, etc. (translation, repeated any number of times)

- we looked at a few multiplications of such elements:
- easy ones:
- <math>H^2</math> = <math>E</math>, <math>R_0{}^2</math> = <math>E</math>, etc.
- <math>T^2</math> x <math>T^3</math> = <math>T^5</math>, etc.

- one hard one: <math>T</math> x <math>R_0</math> = <math>R_{-1}</math>

- easy ones:
- class did the Frieze Group Exploration
**announced an exam for a week from Monday (Oct. 8)**, much like the recent quiz, exercises, explorations:- you can use printed notes for frieze and wallpaper groups
- be able to identify a symmetry group from a pattern
- be able to build a pattern using a given motif and having a given symmetry group
- be able to do identify the elements of a symmetry group (such as from looking at a pattern it describes)
- be able to do multiplication of elements of a rosette symmetry group
- be able to answer questions on angles of polygons such as the 1-6 in Tessellations: Why There Are Only Three Regular Tessellations

- resubmitted projects, exercises, and so on, turned in by Wednesday of next week (Oct. 3) will be returned by Friday; anything after that date won't be available before the exam