Course:Harris, Fall 07: Diary Week 5

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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
      • <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

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>
  • 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