Math 444 Assignment Due Wed 12/5

Problem 1.  Analyze the symmetries of the following patterns.

1. The tessellation by squares of two sizes given out previously.  For this pattern, show that the centers of rotation of the big squares themselves form squares.  Likewise the centers of the small squares also form squares.  If the length of a side of a big square is h, and the side of a small square has length k, what are the smallest distances between (a) two centers of rotation of big squares (b) two centers of rotation of small squares (c) a center of rotation for a big square and a center of rotation for a small square.
2. Look at the brick pavement under the big arch of the Allen Library.  Sketch this on graph paper and analyze the symmetries.

Problem 2.  Explain why the only regular tessellations are made of squares, triangles or hexagons.  The explanation can be a bit informal but should convince a skeptic who knows a little geometry.

Problem 3.  On the Handout of Mirror worksheets, fill out pages 1, 2, 4.  Explain why in such cases of symmetries of a regular n-gon the product of a rotation and a line reflection is a line reflection.  Also, explain in these examples why some pairs of mirrors fill in all the triangles and some do not.

Problem 4.  Draw a point A and a point B.  Construct the centers of rotation of A_72 B_72 and B_72_A_72.  (You can draw the angles with a protractor.)