Assignment 10 (due Monday, 11/23) (70 Points)

Read Bix, Section 9 (carefully) and survey the results in Section 10. Also study Thm. 11.1 (done in class 11/18).

10.1 Triangles, points, and numbers x, y, z. (15 points)

This problem is a sort of converse to 9.1. Start with any triangle ABC you like.

The general question is this. If x, y, z are any numbers so that x + y + z = 1, find a point P so that the x, y, z, numbers coming from the figure in 9.1 are precisely these numbers. In other words, if you are told (x,y, z), find the point P that goes with these values of x, y, z.

In this problem we will no longer assume (implicitly) that the point P in the figure 9.1 is inside the triangle. It can be anywhere on the plane.

Problem: Draw your triangle ABC and the draw and label the following points that correspond to the given numbers (x, y, z).

  1. Point P, where (x,y,z) = (1/3, 1/3, 1/3).
  2. Point Q, where (x,y,z) = (1/2, 1/4, 1/4).
  3. Point R, where (x,y,z) = (1/2, 1/2, 0).
  4. Point S, where (x,y,z) = (0, 1/4, 3/4).
  5. Point T, where (x,y,z) = (0, 1, 0).
  6. Point U, where (x,y,z) = (2, 1, -2).
  7. Point V, where (x,y,z) = (1, 1, -1)).
  8. Point W, where (x,y,z) = (2, 0, -1). (Revised. Original had typo.)

10.2 Finding more Symmetries (15 points)

Do Bix, problem 9.5 for figures i, j, k, o, q. Also indicate the reflection lines and axes of glide reflections. Since these figure are on "graph paper". Write down the translations t and t' in terms of coordinates (e.g., t = (2,3)).

10.3 Glide Reflection Wallpaper (10 points)

Do Bix problems 9.3 and 9.4 together for figures h and q.

10.4 Quad symmetry (10 points)

Bix 9.8.

10.5 Symmetry proof (10 points)

Bix 9.11, 9.12.

10.6 Products of isometries (10 points)

Bix 9.21. This will be done in lab, so you only have to organize your work and turn it in.

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