The Platonic Solids

The counter-earth, Antellus, is intended as a kind of topsy-turvy obverse side of earth. It is, literally, the latent, hidden face of the world we know. On the surface its society is flat and tepid, but the story’s action takes place in the margins and medians peopled by misfits, drolleries, and grotesques. This aspect of Antellus grew in the telling and is likely to continue growing through the sequel. In its original conception, however, the counter-earth was a mathematical conceit with which I entertained myself back when I was a lonely student in a windowless closet of an office at the rear end of a big, chalk-dusty building. Broadly speaking, my dissertation concerned the application of mathematical principles of symmetry to theoretical physics, and my studies included extended forays into general relativity and quantum field theory. But the finite is ever so much more pleasing than the infinite, say I, and I came to be interested in the Platonic solids as a kind of hobby.

A Platonic solid is a convex polyhedron whose faces are congruent regular polygons. Though named after Plato, who made reference to them in his Timaeus, they were not discovered by him. They are five in number. The tetrahedron, the hexahedron (cube), and the dodecahedron were said to have been known to the Pythagoreans, the latter through its resemblance to a certain pyrite crystal that occurs in Italy. The octahedron and icosahedron were discovered by Plato’s contemporary, the Athenian Theatetus; it was Theatetus who also proved that there can be only five such solids. Their construction forms the substance of the last book of Euclid’s Elements. Plato identified the solids with the elements: earth with the cube, air with the octahedron, water with the icosahedron, and fire with the tetrahedron. The fifth solid, the dodecahedron, he described as enveloping the universe.

The Platonic solids have two interesting mathematical properties. The first is duality. Consider the octahedron, which has eight triangular faces. Imagine placing a point in the center of each face. Now connect adjacent points by edges. The cube is thus formed. If we perform the same experiment upon the cube, we obtain the octahedron again. In the same way, the icosahedron (20 faces and 12 vertices) is dual to the dodecahedron (12 faces and 20 vertices), whereas the tetrahedron (4 faces and 4 vertices) is dual to itself.

The second property concerns their symmetry. Take (say) an octahedron. Imagine all the rigid rotations of space about the octahedron’s center of mass that transform the octahedron into itself. It isn’t difficult to see that, because the octahedron has 6 vertices with 4 faces meeting at each vertex, there are 6*4=24 different symmetries, including the one that takes it back to its original position. If we think about the cube in the same way, we see that, because the cube has 8 vertices with 3 faces meeting at each vertex, there are 8*3=24 symmetries. In fact, since the two solids are dual to one another, we know that the set of rotations that preserve one also preserve the other, so we expect this agreement. In just the same way, the number of symmetries of the icosahedron and dodecahedron is 12*5=60=20*3, and the number of symmetries of the tetrahedron is 4*3=12.

It’s interesting to think about what we would get if we took a sphere and a group of transformations (the octahedral group, say), and identified each set of points that get mapped into each other by the group. The resulting space is not very easy to visualize, but it could be described without too much trouble. When I was a student, I got to thinking about what it would be like to inhabit a universe like that, if one were a flatworm, say, or A. Square in Flatland: A Romance of Many Dimensions. The universe we inhabit, they say, is a three-sphere, a three-dimensional analogue of the ordinary sphere. But what if our universe were really a more complicated space resulting from a symmetry of the three-sphere? Or what if it were really a three-sphere, but we could travel from one point to another using a set of symmetries as described above?

Thus Antellus. The counter-earth lies, not beyond the hidden hearth of the solar system, as the Pythagoreans supposed, but at the cosmic antipodes, the dim ultima Thule of the universe.