A curious fact

In Chapter 8 we gave instructions for braiding tetrahedra, cubes, and octahedra. The natural question to ask is: Can we construct the other two Platonic solids by a similar braiding technique? It turns out that we can braid all five of the regular convex solids. In fact, it is easy to verify the following composite statement for the five Platonic solids.

If you make each solid from straight strips of paper fashioned into bands and if you require that all strips on the given model be identical to (or mirror images of) one another, that every edge on the completed model be covered by at least one thickness from the strips, that every face be entirely covered by the strips, and that the same total area from each strip must show on the finished model,

Then you can braid

1. the tetrahedron from 2 strips

2. the hexahedron (cube) from 3 strips

3. the octahedron from 4 strips

4. the icosahedron from 5 strips

5. the dodecahedron from 6 strips.

(The pattern pieces are shown in Figure 9.1.)

We don't have any general explanation for this curious fact, but we will show you how you can easily demonstrate it. This we do by providing you with instructions for constructing the required polyhedra, and once the polyhedra are constructed, you may then verify, simply by looking at them, that they satisfy the conditions of the preceding statement.