If you play role-playing games, there is a good chance you have seen dice with the shapes above. They have 4, 6, 8, 12, and 20 sides. Why did those five shapes get chosen? All the sides are regular polygons and, without the numbers, there is nothing to tell one side from another. This makes the dice as fair as possible. Here is the reason why those shapes were chosen: there are only five shapes in the universe that have those properties! The formal rules for these shapes — to make them fair dice — is that:
- All the faces must be the same regular polygon — with all the sides and angles the same,
- The faces touch one another only at the edges of the polygons, and,
- The same number of faces meet at each corner of the die.
It turns out that there are three more of these shapes, if you allow an infinite number of faces. Intrigued? Read on.
These shapes are called the Platonic solids, after the Greek philosopher Plato. Two different proofs that there are exactly five Platonic solids appear on the Wikipedia page, and one of them uses the same math as one of Occupy Math’s activity posts, a post on the Euler’s map theorem. It is strange that, when you look at solids all of whose faces are the same regular polygon, and which have the same number of faces touching each corner, you get only five possible objects.
The picture above shows the shapes as transparent objects so you can see all the faces. The tetrahedron has four triangular faces, the cube has six square faces, the octahedron has eight triangular faces, the dodecahedron has twelve pentagonal faces, and the icosohedron has twenty triangular faces. The diagram at the top of the article shows how the shapes would look if they were flattened out or cut out of paper.
Why no faces bigger than a pentagon?
The short version of the reason is this. The more sides a regular polygon has, the bigger its angles are. The hexagon’s angles are too big to let it make a die and so are the angles of anything with more than six sides. All the way around a circle is 360° and, at each corner of a platonic solid, at least three shapes must meet. The angles of a regular triangle are 60°, a square 90°, a pentagon 108°, and for a hexagon 120°. If we had three hexagons, then they would use up the whole 360° and so they would have to lie flat, like the picture above. Only pentagons, squares, or triangles can meet so that they actually bend together and make a finite shape. Shapes with more sides than a hexagon have angles that are too large for three of them to meet at all — they would use up more than a circle.
The argument about not using up the circle goes on. To bend inward, the shapes must use less than a whole circle. Only three pentagons can meet at a corner, four of them would use up 432°, more than a circle. Four squares meeting use exactly 360°, so only three squares are allowed, and triangles only have only 60° angles so three, four, or five triangles are allowed, using 180°, 240°, and 300°, respectively. That only those five possibilities are allowed shows that there are at most five Platonic solids — and since we have found five solids, there are exactly five of them.
A really odd connection between the solids
It turns out that the Platonic solids describe one another. The cube and octahedron can be used to build one another and so can the dodecahedron and the icosohedron. The shapes are in pairs that are soulmates. Take a cube, put a point in the center of each face, and then connect two points if the faces they are in the center of touch one another. The shape you get by doing that is an octahedron, as shown at the left. Do the same thing with the faces of an octahedron, and you get a cube. The two shapes each specify one another. This relationship is called topological duality. It turns out that the dodecahedron and icosohedron are also topologically dual to one another and, perhaps strangest of all, the tetrahedron is topologically dual to itself.
This idea of topological duality is not restricted to the Platonic solids. Any solid shape whose faces are polygons can be put through the process outlined in the last paragraph to generate a new solid. Take a cube and cut off the corners; this is a truncated cube, shown at the left. Then you have the polyhedron with six octagonal faces and eight triangular faces. What is its dual polyhedron? Since the truncated cube has three faces meeting at every corner, the dual polyhedron will have triangular faces. Since the truncated cube has 24 corners, the dual polyhedron will have 24 faces. So: twenty-four triangular faces. If you are a good artist, you might try to sketch it.
Infinite versions of Platonic solids
Notice that three hexagons, four squares, and six triangles, meeting at a point, use up all 360° available in a circle. These correspond to all the ways to tile the plane with regular polygons, shown above. The square tiling, or at least part of it, is used quite a bit in games — like a chess board. The hexagonal tiling is often used to give structure to a flat map, for planning purposes. These three structures are close cousins to the Platonic solids — they have all the same properties except for being finite. Since the polygons used up the whole circle, they lie flat and go on forever. It also turns out that the triangular and hexagonal tilings are topologically dual to one another and the square tiling is topologically dual to itself.
Okay, let’s get weird
The Platonic solids and regular tilings are an example of an odd phenomenon — types of objects where only a finite number of examples exist to fill a very general mathematical definition. All of these objects were known to the ancients. If we do not insist on regular tiles, then the situation becomes more complex. The most recently discovered tiling of the plane with pentagons, shown to the left, was found in 2015. Notice that all the pentagons are the same shape, though in different orientations. We are not yet sure that these fifteen different ways to tile the plane with pentagons are all the possibilities, though people are working on proofs.
I hope to see you here again,
Daniel Ashlock,
University of Guelph,
Department of Mathematics and Statistics
Occupy Math 