Course:Harris, Fall 08: Diary Week 5

Mon:

• Class canceled so groups can get to the Cathedral.

Wed:

• Introduction to the mathematical structure known as groups:
  • A group is a set S with
    • a binary operation on S, so that two elements of S can be combined to produce another element of S
      • --the operation must be associative (parentheses don't matter);
    • an "identity" element E: combining any X with E results in X again;
    • for every element $X^{}$ its inverse $X^{-1}$, so that $X^{}X^{-1}=E$ and $X^{-1}X=E$.
  • examples:
    • The integers (positive, negative, and 0) with the operation + :
      • 0 is the identity element.
      • The inverse of 4 is -4, for instance.
    • The symmetry group C4:
      • The elements are those transformations of the plane which preserve a C4-figure:
        • Rotation by 90 degrees counter-clockwise; call this $R$.
      • Double and triple applications of $R{}$, called $R^{2}$ and $R^{3}$.
      • The "null" transformation, i.e., doing nothing to the plane; call this $E$.
      • $R^{}R^{3}=E$, $R^{3}R^{2}=R$, and so on.
      • Thus, $R^{-1}=R^{3}$, $(R^{2})^{-1}=R^{2}$.
    • (When we write two transformations next to one another, such as ST, that means "First do transformation S, then do transformation T.")
    • The symmetry group D4:
      • The same elements as from C4, plus 4 mirror-reflection transformations:
        • $M_{1}$ through $M_{4}$, each its own inverse.
    • The symmetry group C2:
      • As elements, just $E$ and $R'$ (rotation by 180 degrees), which is its own inverse.
• A subgroup of a group is a subset which forms its own group using the same operations.
  • examples:
    • In D4, we can find cyclic subgroups:
      • order 4: C4
      • order 2:
        • {$E$, $R^{2}$}
        • {$E$, $M_{1}$} and so on
• For Symmetry Group for Border Pattern Exploration:
  • elements of MG:
    • an infinite number of rotations (each 180 degrees), ..., $R_{-2}$, $R_{-1}$, $R_{0}$, $R_{1}$, $R_{2}$, ...
    • an infinite number of vertical mirror reflections, ..., $M_{0}$, $M_{1}$, ...
    • a translation $T$ and all its iterations--$T^{2},T^{3}$,...--and its inverse $T^{-1}$ and iterations $T^{-2}$, ...
    • same with a glide-reflection $G$, save that $G^{2}=T$, and so on

Fri:

• We took a closer look at classification of Wallpaper groups:
  • I recommend using the scheme just below the flow chart.
    • First look for rotations.
    • Then for reflection axes.
    • Then for glide-reflection axes (the hardest to find).
  • We looked at two examples of brick patterns:
    • Squares split in two, alternating in orientation; this was p4g.
    • Non-square rectangles split in two, alternating in orientation; this was cmm.
• Groups then had 15 minutes left for the Symmetry Group Border Pattern Exploration.
  • Looking for subgroups in MG:
    • There are two categories of C2 subgroups:
      • {E, $R_{n}$}, for any rotation $R_{n}$.
      • Similarly, using any of the vertical mirror reflections.
    • There is also a 11 subgroup, formed by all the iterations of $T$ (translation) and its inverse $T^{-1}$.
• No Exercises for Monday; but try the Regular Triangle Symmetry Group Exploration, that we didn't have time to get to in class.