Course:Harris, Fall 08: Diary Week 3

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:
    • $R_{1}^{}$ = 90-degree rotation, counter-clockwise, with center Q1
    • $R_{1}^{-1}$ = the inverse rotation: 90 degrees clockwise = 270 degrees counter-clockwise with center Q1
    • $R_{1}^{2}$ = applying $R_{1}^{}$ twice, i.e., 180 degrees with center Q1
    • similarly for center Q2: $R_{2}^{}$, $R_{2}^{-1}$, $R_{2}^{2}$
  • Successive application of $R_{1}^{}$, $R_{2}^{-1}$, $R_{1}^{}$, $R_{2}^{-1}$, 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.