# Course:Harris, Fall 08: Diary Week 3

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- 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.

- if you rotate once by given amount, then the

- Explained that n-fold rotation means that if we label the vertices of the figure, then
- 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.

- Consider a square grid, with two center points of 4-fold rotation, 3 squares apart (as in the exercise):

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.

- We considered 90-degree rotations from center points Q1 and Q2:
- 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.

- an extra-credit project: Show

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.