Math 444 Assignment Due Wed 11/28

Reading Assignment. Start reading Brown Chapter 2, with the goal of finishing it this week.

Notation note: On this web page, something like R_m should be read as "R subscript m".

Change of Problem. The problem 2 originally on this assignment sheet was replaced by a new one, given below. The original problem will be part of Friday's assignment.

1. Suppose m and n are lines; denote the corresponding line reflections by M and N.  Prove that MN = NM if and only if m is perpendicular to n.
2. On the two sheets handed out in class (here is a link to the c-shaped octagonal one and also a link to the one with squares), study glide reflections. In each case do the following.

• First, pick two shapes in the pattern and find a glide reflection that takes one to the other. Do this for several pairs of shapes.
• Second, find any glide reflections that are symmetries of the whole pattern.

3. Answer both.

a.  State which triangles have point symmetry and prove it.

b.  State which quadrilaterals have point symmetry and prove it.

4. Given a point A and a line m, state and prove exactly which isometry is R__m A_₁₈₀. 

Original Problem 2, now due as Part of Friday's assignment.

2. Draw 2 points A and B on a sheet of paper.  Then we denote the rotation with center A by 180 degrees and the rotation with center B by 60 degrees as the rotations A₁₈₀ and B ₆₀. Then the composition T = A₁₈₀ B ₆₀ is a rotation also. Construct the center of rotation C for T and also construct the angle of rotation.  Then do the same for the center D of S = B ₆₀ A₁₈₀. (Directions for doing this are found in Brown 2.4, and also will be given in class.)