# Course:Harris, Fall 08: Diary Week 6

Mon:

• We looked at problems in identifying Wallpaper Symmetry Groups:
• First look for rotations.
• Then for reflections.
• Then for glide-reflections; these can be difficult to spot.
• We reviewed what it means to have a subgroup of a group of symmetries.
• In the Frieze Groups, the cyclic subgroups are all of the form {[itex]E[/itex],[itex]X[/itex]}, where
• [itex]E[/itex] is the identity transformation.
• [itex]X[/itex] is a reflection or a rotation; in either case, [itex]X^2[/itex] = [itex]E[/itex], closing off the subgroup.
• The question was asked, why do I specify some symmetries with superscripts, some with subscripts?
• Superscripts indicate multiplication: [itex]TT = T^2[/itex], i.e., [itex]T^2[/itex] indicates doing [itex]T[/itex] twice.
• Subscripts are just labels: [itex]R_1[/itex] is a different rotation from [itex]R_2[/itex], and there is not necessarily any special significance to the choice of label.
• We spent the remainder of the period taking our first look at multiplying transformations in a complex manner, taking our example from D4.
• We looked at two transformations within D4:
• [itex]R[/itex] (rotation by 90 degrees)
• [itex]M_1[/itex] (reflection across one of the reflecting lines)
• We chose to look at D4 as exemplified in a square; [itex]M_1[/itex] was reflection across the NE-SW diagonal line.
• We first looked at [itex]RM_1[/itex],i.e., if we do first [itex]R[/itex] and then [itex]M_1[/itex], what is the resulting transformation on the plane? It's got to be one of the other symmetries of the square, i.e., some other element of D4.
• We found [itex]RM_1 = M_4[/itex], reflection across the E-W line.
• We then looked at [itex]M_1R[/itex], i.e., doing first [itex]M_1[/itex] and then [itex]R[/itex].
• We found [itex]M_1R[/itex] = [itex]M{}_2[/itex], reflection across the N-S line.
• Note: [itex]RM_1[/itex] and [itex]M_1R[/itex] are not equal! This is non-commutative "multiplication".

Wed:

• Groups worked on the D4 symmetry group (question 1 from Regular Triangle Symmetry Group Exploration), taking most of the period.
• Hints for working on these kinds of problems:
• [itex]E[/itex] times anything (and anything times [itex]E[/itex]) is very easy.
• Multiplication of [itex]R^n[/itex][itex]R^m[/itex] is always easy.
• Any column (and any row) must contain all the elements of the group.
• Orientation:
• This is
• preserved by any kind of rotation,
• reversed by any kind of reflection.
• Thus (with [itex]R^i[/itex] being any rotation and [itex]M_j[/itex] any reflection),
• [itex]R^i[/itex][itex]M_j[/itex] must be an [itex]M_k[/itex] (because orientation is preserved, then reversed: net, reversed)
• [itex]M_i[/itex][itex]R^j[/itex] must be an [itex]M_k[/itex] (because orientation is reversed, then preserved: net reversed)
• [itex]M_i[/itex][itex]M_j[/itex] must be an [itex]R^k[/itex] (because orientation is reversed, then reversed: net, preserved)
• Any progress towards the D3 group (question 8) well be added in as extra credit.
• Exam 1 will probably be next Wednesday, Oct. 8. Included:
• Identification of symmetry groups as per quizzes and the Cathedral project.
• Multiplying group elements, as per today's Exploration.
• Ideas from the next Exploration, on Why There Are Only Three Regular Tessellations.

Fri:

• Exam 1 is moved to Friday of next week, Oct. 10 (I'm planning on getting a substitute instructor for that day, as I'll be heading to the airport for an afternoon flight).
• 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 or why there are only three regular tessellations, as in Tessellations: Why There Are Only Three Regular Tessellations.
• We looked at the angle sum for any n-gon (n-sided polygon):
• Any polygon can be subdivided into triangles, with
• all triangle edges going between vertices of the polygon and
• no triangles overlapping.
• For an n-gon, it takes n-2 triangles to subdivide it:
• Surely this is true for n = 3 (i.e., a triangle has 1 triangle "subdividing" it).
• Suppose this formula were true for all polygons up to size N.
• Then, what about for n = N+1? Given an (N+1)-gon, we can slice off two adjacent edges, say, A-B-C, replacing them with A-C, thus giving us an N-gon (sides AB and BC replaced by AC, so 1 fewer edge).
• The N-gon can be subdivided into N-2 triangles (because we're supposing the formula true for n = N).
• Then adding back in triangle ABC, we have N -2+1 = N-1 triangles subdividing the original (N+1)-gon.
• That's the number we wanted, since N+1-2 = N-1.
• Thus, if the formula holds for n = N, it also holds for n = N+1.
• Thus, by the Principle of Induction, the formula holds for all n: Any n-gon can be subdivided into n-2 triangles.
• The sum of all the angles of all the n-2 subdividing triangles, adds up to the angle-sum for the n-gon (we need the triangles to be non-overlapping for this, and also that the triangle edges go between vertices of the n-gon).
• Thus, the angle-sum for the n-gon is n-2 times the angle-sum of a triangle, i.e., (n-2)180 degrees.
• We looked, then at the angle-formula for a regular n-gon:
• Since an n-gon has n vertices, there are n angles.
• We know the angle-sum (the sum of all those n angles) is (n-2)180.
• We know all n angles, in a regular n-gon, are the same.
• Thus, each of those angles must be (n-2)180/n.
• We changed the Exercises for Monday to questions 7-10 in Tessellations: Why There Are Only Three Regular Tessellations.