Course:Harris, Fall 08: Diary Week 5

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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 XX − 1 = E and X − 1X = 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 R2 and R3.
        • The "null" transformation, i.e., doing nothing to the plane; call this E.
        • RR3 = E, R3R2 = R, and so on.
        • Thus, R − 1 = R3, (R2) − 1 = R2.
      • (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:
          • M1 through M4, 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, R2}
          • {E, M1} 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, R0, R1, R2, ...
      • an infinite number of vertical mirror reflections, ..., M0, M1, ...
      • a translation T and all its iterations--T2,T3,...--and its inverse T − 1 and iterations T − 2, ...
      • same with a glide-reflection G, save that G2 = 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, Rn}, for any rotation Rn.
        • 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.
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