Homework #5
Due Tuesday, October 27

1. Find all the cube roots of i.
2. Draw any non-convex quadrilateral such that no two sides are the same length. Draw an ``Escher-like'' tiling of the plane by modifying the sides of this quadrilateral in the manner that we discussed in class. You may use Geometer's Sketchpad to do this if you wish.
3. Explain how you can be absolutely certain that we have found all 21 possible planar clusters. Some suggestions: What are the possible numbers of polygons that can be in a cluster? For each possible number, what are the possible numbers of triangles that can be in a cluster? For each possible number of triangles, what are the possible numbers of squares? Etc.
4. Consider two points and in space.
   1. What are the coordinates of the midpoint of the segment ?
   2. Suppose we wish to divide the segment by a point C such that AC:CB=3:2. Determine the coordinates of C.
5. Determine coordinates of the vertices of the polyhedron determined by the space cluster 3-4-3-4. Use Maple to draw it. Print out the result.
6. Read the article on imaginary numbers and the two articles on tilings that are on reserve in the Education Library.

You should have already done this:

• Make good drawings of the possible tilings by planar clusters.
• Find samples of border wallpaper and regular wallpaper with interesting repeating patterns.

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