t together three, four, or five of them at a vertex, and these correspond to the tetrahedron, the octahedron, and the icosahedron. Each interior angle of a square is 90, so we can fit only three of them together at each vertex, giving us a cube. The interior angles of the regular pentagon are 108, so again we can fit only three together at a vertex, giving us the dodecahedron. That makes five regular polyhedra. However, what would happen if we had a six-sided figure? Well, its interior angles are 120, so if we fit three of them together at a vertex the angles add up to 360, and therefore they lie flat. For this reason we cannot use hexagons to make a Platonic solid. In addition, obviously, no polygon with more than six sides can be used either, because the interior angles just keep getting larger. The Greeks, who had to find religious truth in mathematics, found the idea of exactly five Platonic solids very compelling. The philosopher Plato concluded that they must be the fundamental building blocks of nature, and assigned to them what he believed to be the essential elements of the universe. He followed the earlier philosopher Empedocles in assigning fire to the tetrahedron, earth to the cube, air to the octahedron, and water to the icosahedron. To the dodecahedron, Plato assigned the element cosmos, reasoning that, since it was so different from the others, because of its pentagonal faces, it must be what the stars and planets are composed of. Although this might seem odd to us, these were really very powerful ideas, and led to real knowledge. As late as the 16th century, Johannes Kepler was applying a similar intuition to attempt to explain the motion of the planets. Early in his life, he concluded that the distances of the orbits, which he assumed were circular, were related to the Platonic solids in their proportions. Only later in his life, after his friend the great astronomer Tycho Brahe gave him a his collection of astronomi...
