Course:Harris, Fall 07: Diary Week 5

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)
    • $R^{2}$ = 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

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):
    • cylinder
    • Möbius strip
    • torus (donut)
    • Klein bottle
    • projective plane
  • 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: $E$, $M1$, $M2$, $M3$, $M4$, $R$, $R^{2}$, $R^{3}$
  • the easy multiplications: $R$ x $R^{2}$, 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)
  • $V_{0}$, $V_{1}$, $V_{2}$, etc. (many vertical reflections)
  • $R_{0}$, $R_{1}$, $R_{2}$, etc. (many 180-degree rotations)
  • $T$, $T^{2}$, $T^{3}$, etc. (translation, repeated any number of times)
• we looked at a few multiplications of such elements:
  • easy ones:
    • $H^{2}$ = $E$, $R_{0}{}^{2}$ = $E$, etc.
    • $T^{2}$ x $T^{3}$ = $T^{5}$, etc.
  • one hard one: $T$ x $R_{0}$ = $R_{-1}$
• 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