We are all familiar with dice of various kinds. With fair dice, each number will come up with equal probability, regardless of how the die is rolled. This requires a degree of symmetry – we want a die to be a polyhedron where all the faces are equivalent. The obvious candidates are therefore the five Platonic solids, in which not only the faces, but also the edges and vertices are equivalent. The Platonic solids give us the common d4, d6, d8, d12, and d20 dice:
Five Platonic dice (d4, d6, d8, d12, d20) and two pentagonal trapezohedra (d10) – photo by “Copat”
However, the Platonic solids are more symmetrical than necessary for the job. A tetragonal disphenoid, for example, makes a very good d4:
A tetragonal disphenoid makes an alternative d4 – photo by “Traitor”
What is required is that a die be isohedral (also called “face-transitive”). Each face should be equivalent. Specifically, for any numbers A and B, given the die with A on top, there should be a series of rotations and reflections that make the die look like the starting position, but with B on top. This rules out shapes like the gyroelongated square bipyramid, where all the faces are equilateral triangles, but the triangles are not equivalent (the “end” triangles differ from the “middle” triangles):
A gyroelongated square bipyramid does not make a fair die – photo by Andrew Kepert
We also want a die to be convex, so that it can land on its faces. Stellated polyhedra are excluded:
Trapezohedra satisfy this “convex and isohedral” rule, and the pentagonal trapezohedron is commonly used as a d10 (see the picture above). Trapezohedra work best with 10, 14, 18, … sides, since then pairs of faces can be parallel, and there can be an unambiguous “top” number. The cube can be seen as a special case of a trapezohedron.
For 12, 16, 20, …. sides, bipyramids make good dice (the octahedron is a special case of a bipyramid):
A bipyramid makes a good d16 – photo by “Traitor”
These are not the only shapes satisfying the definition, however. The 13 Catalan solids also satisfy it, and some of them make good candidates for dice. For example, the deltoidal icositetrahedron and the tetrakis hexahedron are both good candidates for d24:
The deltoidal icositetrahedron and tetrakis hexahedron are alternatives for d24 – photos by Jacqueline de Swart (left) and “Traitor” (right)
Some Catalan solids, like the pentagonal icositetrahedron, are unsuitable as dice because there is no unambiguous “top” number. On the other hand, there are some additional variations that are isohedral, like the hexakis tetrahedron.
For more on this subject, see Alea Kybos’ impressive dice page.