# 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
• [itex]E[/itex] = identity/neutral element (no change)
• [itex]R[/itex] = 120-degree rotation counterclockwise (CCW is default meaning of positive rotation)
• [itex]R^2[/itex] = doing [itex]R[/itex] twice
• nothing else, as repeating [itex]R[/itex] any number of times (including backward) just results in one of the above
• D2 (symmetries of a rectange), consisting of
• [itex]E[/itex] = identity
• [itex]R[/itex] = 180-degree rotation
• [itex]M1[/itex] = reflection across vertical axis
• [itex]M2[/itex] = 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):
• 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: [itex]E[/itex], [itex]M1[/itex], [itex]M2[/itex], [itex]M3[/itex], [itex]M4[/itex], [itex]R[/itex], [itex]R^2[/itex], [itex]R^3[/itex]
• the easy multiplications: [itex]R[/itex] x [itex]R^2[/itex], etc.
• a couple of the "interesting" multiplications: [itex]M1[/itex] x [itex]M2[/itex], [itex]M2[/itex] x [itex]M1[/itex]
• 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:
• [itex]E[/itex]
• [itex]H[/itex] (horizantal reflection)
• [itex]V_0[/itex], [itex]V_1[/itex], [itex]V_2[/itex], etc. (many vertical reflections)
• [itex]R_0[/itex], [itex]R_1[/itex], [itex]R_2[/itex], etc. (many 180-degree rotations)
• [itex]T[/itex], [itex]T^2[/itex], [itex]T^3[/itex], etc. (translation, repeated any number of times)
• we looked at a few multiplications of such elements:
• easy ones:
• [itex]H^2[/itex] = [itex]E[/itex], [itex]R_0{}^2[/itex] = [itex]E[/itex], etc.
• [itex]T^2[/itex] x [itex]T^3[/itex] = [itex]T^5[/itex], etc.
• one hard one: [itex]T[/itex] x [itex]R_0[/itex] = [itex]R_{-1}[/itex]
• 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