# 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 [itex]X^{}[/itex] its inverse [itex]X^{-1}[/itex], so that [itex]X^{}X^{-1} = E[/itex] and [itex]X^{-1}X = E[/itex].
• 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 [itex]R[/itex].
• Double and triple applications of [itex]R{}[/itex], called [itex]R^2[/itex] and [itex]R^3[/itex].
• The "null" transformation, i.e., doing nothing to the plane; call this [itex]E[/itex].
• [itex]R^{}R^{3} = E[/itex], [itex]R^{3}R^{2} = R[/itex], and so on.
• Thus, [itex]R^{-1} = R^3[/itex], [itex](R^2)^{-1} = R^2[/itex].
• (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:
• [itex]M_1[/itex] through [itex]M_4[/itex], each its own inverse.
• The symmetry group C2:
• As elements, just [itex]E[/itex] and [itex]R'[/itex] (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:
• {[itex]E[/itex], [itex]R^{2}[/itex]}
• {[itex]E[/itex], [itex]M_1[/itex]} and so on
• For Symmetry Group for Border Pattern Exploration:
• elements of MG:
• an infinite number of rotations (each 180 degrees), ..., [itex]R_{-2}[/itex], [itex]R_{-1}[/itex], [itex]R_0[/itex], [itex]R_1[/itex], [itex]R_2[/itex], ...
• an infinite number of vertical mirror reflections, ..., [itex]M_0[/itex], [itex]M_1[/itex], ...
• a translation [itex]T[/itex] and all its iterations--[itex]T^2, T^3[/itex],...--and its inverse [itex]T^{-1}[/itex] and iterations [itex]T^{-2}[/itex], ...
• same with a glide-reflection [itex]G[/itex], save that [itex]G^2 = T[/itex], 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, [itex]R_n[/itex]}, for any rotation [itex]R_n[/itex].
• Similarly, using any of the vertical mirror reflections.
• There is also a 11 subgroup, formed by all the iterations of [itex]T[/itex] (translation) and its inverse [itex]T^{-1}[/itex].
• No Exercises for Monday; but try the Regular Triangle Symmetry Group Exploration, that we didn't have time to get to in class.