# Course:Harris, Fall 08: Diary Week 3

Mon:

• 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:
• [itex]R_1^{}[/itex] = 90-degree rotation, counter-clockwise, with center Q1
• [itex]R_1^{-1}[/itex] = the inverse rotation: 90 degrees clockwise = 270 degrees counter-clockwise with center Q1
• [itex]R_1^2[/itex] = applying [itex]R_1^{}[/itex] twice, i.e., 180 degrees with center Q1
• similarly for center Q2: [itex]R_2^{}[/itex], [itex]R_2^{-1}[/itex], [itex]R_2^{2}[/itex]
• Successive application of [itex]R_1^{}[/itex], [itex]R_2^{-1}[/itex], [itex]R_1^{}[/itex], [itex]R_2^{-1}[/itex], 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.