Assignment 10A (Due Wednesday, 12/4)

Read Berele-Goldman, Sections 8.1, 8.3, 9.1, 9.2

Correction to table in 10-1 is in red below (F270).

Answers to problems 10-1 and 10-2 are now linked.

10-1: This figure is made of squares, with the centers of the squares labeled J, K, L, M.

The goal of this exercise is to begin with S = A90 and T = B90 and to write rotations with centers at the other points as products of S and T.  For a number of cases you are asked to do this explicitly. In this table,

a)      Fill in column 2 by writing the given rotation as a product of two rotations already known from above as products of S's and T's.

b)      Fill in column 3 by writing the given rotation as a product of S's and T's.

c)      Explain below the table what are all the centers of rotations that are products of S's and T's and how you would prove this.

d)      Also, tell what are the translations that are products of S's and T's.  In particular, is the translation that takes A to B such a product? If so, what is the product?

10-2: Draw a point A and a line m.  Let M be reflection in m.

a)      Explain clearly how you know that for any rotation S with center A, the product MS is a glide reflection. 

b)      Then draw A and m and construct the invariant lines of the glide reflections MA₉₀, MA₁₈₀, MA₂₇₀. 

c)      Indicate the glide vector (the translation vector used in defining the glide reflection) of each of these isometries.

Problem 9-4 Second Try (Napoleon's Theorem)

Turn in Problem 9-4 along with Assignment 10A for full credit.