Course:Harris, Fall 08: Diary Week 3

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Mon:

  • Spent about 20 minutes (too long!) going over the answers to Quadrilaterals Exploration.
    • Perhaps most useful bit was why parallelograms must be convex.
  • Groups finished Symmetric Stars/Polygons Exploration, some taking ~30 minutes.
    • Explained that n-fold rotation means that if we label the vertices of the figure, then
      • if you rotate once by given amount, then the unlabeled figure is brought onto itself, and
      • if you rotate n times by that same amount, then the figure is brought back onto itself, with all labels coming back to themselves also.
  • Looked at Chinese Boy design:
    • There are multiple center points for D3 symmetries (rotation + reflection).
    • There are multiple center points for C3 symmetries (rotation without reflection).
  • Three classes of designs:
    • compact
    • extending infinitely in one direction
    • extending infinitely in multiple directions
  • If a design has multiple center points for rotational symmetry, then is it possible for the design to be compact? (#15 of Rosette Exercises)
    • Consider a square grid, with two center points of 4-fold rotation, 3 squares apart (as in the exercise):
      • If we do a rotation by 90 degrees from the left center-point, where does the marked square (in the exercise) go to? --homework!
      • To be finished in class next time.


Wed:

  • I observed (just general remark) that in order for this class to function, it's important that students do the readings and any other preparations beforehand.
  • Symmetries are rigid transformations of the plane (or the sphere or "hyperbolic space"...)
    • rotations of the plane about a center point
    • reflections of the plane across an axis
    • translations of the plane in a specific direction by a specific amount
    • glide-reflections: translations of the plane in a specific direction by a specific amount combined with a reflection across an axis aligned with the translation direction
  • exercise #15 of Polygon Exercises: If the symmetries of a figure include rotations from two centers points, can the figure be compact?
    • We considered 90-degree rotations from center points Q1 and Q2:
      • <math>R_1^{}</math> = 90-degree rotation, counter-clockwise, with center Q1
      • <math>R_1^{-1}</math> = the inverse rotation: 90 degrees clockwise = 270 degrees counter-clockwise with center Q1
      • <math>R_1^2</math> = applying <math>R_1^{}</math> twice, i.e., 180 degrees with center Q1
      • similarly for center Q2: <math>R_2^{}</math>, <math>R_2^{-1}</math>, <math>R_2^{2}</math>
    • Successive application of <math>R_1^{}</math>, <math>R_2^{-1}</math>, <math>R_1^{}</math>, <math>R_2^{-1}</math>, etc., moves a certain square infinitely far up and right and left.
    • That means that if a figure contains that square and has these symmetries, then all those images of that square must also be in the figure, so the figure is not compact.
  • A compact figure can have only these symmetries:
    • rotations from one center
    • reflections (all axes must pass through one point, same as the rotation center)---we haven't proved this in class, though.
      • an extra-credit project: Show why a compact figure cannot have
        • two parallel reflection axes or
        • three reflection axes not passing through a common point.


Fri:

  • There are two goals of this study of symmetry groups:
    • Given a figure, find (and name) its symmetry group.
    • Given a symmetry group and a particular motif, construct a figure using repetitions of that motif so that the figure has that symmetry group.
  • There will be a short quiz next week (probably Wed.) asking you to do one or another of those tasks for:
    • rosettes and rosette groups;
    • friezes and frieze groups.
  • The Cathedral Group Project will be week after next, on Monday; see the schedule for additional details.
  • I handed out the alternative description of the frieze symmetry groups.
  • Groups did the Tessellation Basics Exploration, apparently without much problem (a little discussion on what it means for vertices to be "the same").
  • Groups started the Border Patterns Exploration, to be completed Monday.