## Math 110-02, Fall 2008Polyhedra Exercises

This is the last week of classes for the term. The final reading from Symmetry, Shape, and Space is Sections 7.1 and 7.2, excluding the part about Volume Formulae on pages 212 to 214. We will work on this material in class on Monday and Wednesday, and it will be included on the final exam, but there is no homework assignment on this material.

The final homework, a short assignment on mathematical infinity, is due in class on Wednesday.

For Friday's class, you should read and be prepared to discuss the last two chapters in The Math Instinct, Chapters 12 and 13, pages 239 to 264.

The final exam is at 8:30 AM on Thursday, December 18. The final project is also due at that time, but you can certainly turn it in earlier if you want. If you want to get your project back, you should do so during the first few weeks of the Winter semester; after that, I will throw them out.

### The Platonic Solids

For class on Monday, December 8, you will work in groups to build several regular polyhedra. You should break into groups of three. Each group of three will build a model of each of the five regular polyhedra and will attempt to build each of the five irregular convex deltahedra. (The deltahedra are defined on pages 222 and 223.) Each group will produce just one model of each of these. We'll collect them at the front of the room and see how much you can get done. As you make the models, you should fill out the table on the back of this sheet

For the dodecahedron, each group should get from me a printout of a dodecahedral net. Cut out the net (in one piece), then fold it and tape it together to make a dodecahedron. (To keep it neat, try putting the tape on the inside.)

For the tetrahedron, octahedron, and icosahedron, you should use printouts of a triangular tiling. For each polyhedron, cut a net out of the tiling -- all in one piece if possible -- then fold and tape the net to form the polyhedron. A tetrahedron is made from four triangles, an octahedron from eight, and an icosahedron from twenty. A net for the tetrahedron is easy. An octahedron is also easy if you remember that it looks like two square pyramids connected by their bases. A square pyramid could be made from four triangles connected like wedges of a pie. An icosahedron is harder: it consists of two pentagonal pyramids with their bases connected by a strip of ten triangles.

For the cube, you are on your own. It should be easy to draw a net for the cube, cut it out, and assemble it

### Deltahedra

As defined in the book, a deltahedron is a polyhedron whose sides are all equilateral triangles of the same size. The regular deltahedra are the tetrahedron, the octahedron, and the icosahedron. But there five additional convex deltahedra, with 6, 10, 12, 14, and 16 sides. Try to find them and build models. As a hint, you'll find that several of them contain triangular, square, or pentagonal pyramids as components. Some of these are pretty easy to find; the twelve-sided convex deltahedron is the most difficult to discover. (Some of the descriptions in Section 7.2, Exercise 26, might be helpful.)

### Table of Euler Characteristics

Polyhedron Number
of vertices
(v)
Number
of edges
(e)
Number
of faces
(f)
Euler
Characteristic
(χ = v - e + f)
tetrahedron
cube
octahedron
dodecahedron
icosahedron
6-sided deltahedron
10-sided deltahedron
12-sided deltahedron
14-sided deltahedron
16-sided deltahedron