Course:Harris, Fall 08: Diary Week 5

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  • Class canceled so groups can get to the Cathedral.


  • 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 <math>X^{}</math> its inverse <math>X^{-1}</math>, so that <math>X^{}X^{-1} = E</math> and <math>X^{-1}X = E</math>.
    • 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 <math>R</math>.
        • Double and triple applications of <math>R{}</math>, called <math>R^2</math> and <math>R^3</math>.
        • The "null" transformation, i.e., doing nothing to the plane; call this <math>E</math>.
        • <math>R^{}R^{3} = E</math>, <math>R^{3}R^{2} = R</math>, and so on.
        • Thus, <math>R^{-1} = R^3</math>, <math>(R^2)^{-1} = R^2</math>.
      • (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:
          • <math>M_1</math> through <math>M_4</math>, each its own inverse.
      • The symmetry group C2:
        • As elements, just <math>E</math> and <math>R'</math> (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:
          • {<math>E</math>, <math>R^{2}</math>}
          • {<math>E</math>, <math>M_1</math>} and so on
  • For Symmetry Group for Border Pattern Exploration:
    • elements of MG:
      • an infinite number of rotations (each 180 degrees), ..., <math>R_{-2}</math>, <math>R_{-1}</math>, <math>R_0</math>, <math>R_1</math>, <math>R_2</math>, ...
      • an infinite number of vertical mirror reflections, ..., <math>M_0</math>, <math>M_1</math>, ...
      • a translation <math>T</math> and all its iterations--<math>T^2, T^3</math>,...--and its inverse <math>T^{-1}</math> and iterations <math>T^{-2}</math>, ...
      • same with a glide-reflection <math>G</math>, save that <math>G^2 = T</math>, and so on


  • 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, <math>R_n</math>}, for any rotation <math>R_n</math>.
        • Similarly, using any of the vertical mirror reflections.
      • There is also a 11 subgroup, formed by all the iterations of <math>T</math> (translation) and its inverse <math>T^{-1}</math>.
  • No Exercises for Monday; but try the Regular Triangle Symmetry Group Exploration, that we didn't have time to get to in class.