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Question of the Day

1. What is the most symmetric figure that you can draw in the plane?
2. What if you could use only straight lines?
3. Now what is the most symmetric figure in space that you can imagine?
4. What if you could only use flat sides?

Key Concepts

1. There are only 5 regular solids.
2. By counting edges, faces, and vertices, patterns emerge that lead to the idea of duality.
3. The Golden Ratio appears in surprising places.

Vocabulary

1. The Platonic solids, also called the regular solids or regular polyhedra, are convex polyhedra with equivalent faces composed of congruent convex regular polygons. There are exactly five such solids. (from mathworld.com)

Mathematical Ideas

Buiding the solids

The manipulative kit comes with materials to build the solids. Build your own models of the platonic solids.

Examining the solids

Fill in the entries in the following table:

Are there relationships among the values in the table? For instance, why should the number of vertices times the number of edges per vertex be twice the number of edges? (We will encounter this kind of reasoning again in Session C on Counting.) Are there “reciprocal” or “complementary” relationships among the number of vertices and the number of faces between different rows in the table?

Duality

Visualize yourself inside a cube (for instance a roughly cubical room like a classroom). Put a vertex in the center of each face, and then connect the vertices with edges. What do you get? What happens if you do the same with a tetrahedron.

The Golden Rectangle in a Dodecahedron

Using the dodecahedron you constructed from straws, stand or balance the dodecahedron on an edge. You should have a parallel edge at the “top” of this balancing dodecahedron. Imagine a rectangle which goes through the middle of the dodecahedron with these two edges as the short edges of the rectangle. The length of the rectangle will be the “height” of your balancing dodecahedron. Measure and record both the length of the short side and the long side of this rectangle. Then take the ratio of the long side length to the short side. If you have a group, average all the ratios. How close is your result to the Golden Ratio? (When we did this exercise at the February 4 workshop, we had an average of 1.68, agreement with Golden Ratio good to two decimal places, probably all you can expect from a soda strawn and pipe cleaner construction and a plastic ruler.)

This section is adapted from: Instructor Resources and Adjunct Guide for the second edition of The Heart of Mathematics by E. Burger, M. Starbird, and D. Bergstrand.

Problems to Work for Understanding

1. Solidifying Ideas: pages 284-285, 4,6,7,8,10
2. New Ideas: page 286, 2,3
3. Habits of Mind: page 286, 2

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