Course:Harris, Fall 08: Diary Week 6

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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 {<math>E</math>,<math>X</math>}, where
      • <math>E</math> is the identity transformation.
      • <math>X</math> is a reflection or a rotation; in either case, <math>X^2</math> = <math>E</math>, closing off the subgroup.
  • The question was asked, why do I specify some symmetries with superscripts, some with subscripts?
    • Superscripts indicate multiplication: <math>TT = T^2</math>, i.e., <math>T^2</math> indicates doing <math>T</math> twice.
    • Subscripts are just labels: <math>R_1</math> is a different rotation from <math>R_2</math>, 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:
      • <math>R</math> (rotation by 90 degrees)
      • <math>M_1</math> (reflection across one of the reflecting lines)
        • We chose to look at D4 as exemplified in a square; <math>M_1</math> was reflection across the NE-SW diagonal line.
    • We first looked at <math>RM_1</math>,i.e., if we do first <math>R</math> and then <math>M_1</math>, 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 <math>RM_1 = M_4</math>, reflection across the E-W line.
    • We then looked at <math>M_1R</math>, i.e., doing first <math>M_1</math> and then <math>R</math>.
      • We found <math>M_1R</math> = <math>M{}_2</math>, reflection across the N-S line.
    • Note: <math>RM_1</math> and <math>M_1R</math> 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:
      • <math>E</math> times anything (and anything times <math>E</math>) is very easy.
      • Multiplication of <math>R^n</math><math>R^m</math> 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 <math>R^i</math> being any rotation and <math>M_j</math> any reflection),
          • <math>R^i</math><math>M_j</math> must be an <math>M_k</math> (because orientation is preserved, then reversed: net, reversed)
          • <math>M_i</math><math>R^j</math> must be an <math>M_k</math> (because orientation is reversed, then preserved: net reversed)
          • <math>M_i</math><math>M_j</math> must be an <math>R^k</math> (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.