Course:Harris, Fall 07: Diary Week 5
From EscherMath
Mon:
- 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
- E = identity/neutral element (no change)
- R = 120-degree rotation counterclockwise (CCW is default meaning of positive rotation)
- R2 = doing R twice
- nothing else, as repeating R any number of times (including backward) just results in one of the above
- D2 (symmetries of a rectange), consisting of
- E = identity
- R = 180-degree rotation
- M1 = reflection across vertical axis
- M2 = 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: E, M1, M2, M3, M4, R, R2, R3
- the easy multiplications: R x R2, etc.
- a couple of the "interesting" multiplications: M1 x M2, M2 x M1
- 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:
- E
- H (horizantal reflection)
- V0, V1, V2, etc. (many vertical reflections)
- R0, R1, R2, etc. (many 180-degree rotations)
- T, T2, T3, etc. (translation, repeated any number of times)
- we looked at a few multiplications of such elements:
- easy ones:
- H2 = E, R02 = E, etc.
- T2 x T3 = T5, etc.
- one hard one: T x R0 = R − 1
- 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
