A Platonic solid is as regular as a three-dimensional shape can be: every face is the same regular polygon, the same number of faces meet at every corner, and every corner sits on one sphere. Remarkably, only five such shapes exist, and the reason is a matter of simple arithmetic — the angles meeting at a corner have to add up to less than a full turn, which leaves room for three, four or five triangles, three squares, or three pentagons, and nothing else. The proof closes the list for good, which is why no sixth one has ever been found.
The Greeks knew all five and Plato gave them their lasting fame by pairing them with the elements — fire, earth, air, water and the heavens — in a scheme that was wrong as physics but memorable as a picture. Euclid closed the Elements with a construction of each one and a proof that the list is complete, and the shapes have kept turning up ever since: Kepler built an early model of the solar system by nesting them inside one another, and when that failed he went on to the elliptical orbits that made his name.
They also come in pairs. Put a point at the centre of each face of a cube and join the neighbours, and an octahedron appears; do the same to the octahedron and the cube comes back. The dodecahedron and icosahedron trade places the same way, and the tetrahedron is its own partner. Beyond the mathematics they are thoroughly practical: the cube for dice and packing, the icosahedron for geodesic domes, for the shells of many viruses and for the twenty-sided dice of tabletop games, and all five for the standard set that role-playing games have used for half a century.