lpetrich
Contributor
We are all familiar with cubical dice and we sometimes use coins as dice, and many players of tabletop role-playing games use dice with a variety of different shapes. For n faces, they call their dice d<n>. Thus, an ordinary cubical die is d6 and a coin is d2.
I will now derive what possible sorts of fair dice there are, a fair die being one whose faces act alike. Here are some names for such dice and similar polyhedra:
The arguments that I will be using I will generalize to find plane tilings. Continuing with gaming metaphors, these are isohedral dartboards, fair ones with lookalike faces. List of convex uniform tilings has a list of them.
For a hyperbolic surface, one can also construct tilings, Uniform tilings in hyperbolic plane. These ones have the remaining parameter values, including faces with infinite numbers of vertices and vertices with an infinite number of faces. The hyperbolic plane is usually displayed as a disk, with most of the plane scrunched up near the edge of the disk. So one puts one end of an infinite-vertex face at the edge and one also puts an infinite-face vertex there.
There is an interesting interrelationship that some polyhedra have: duality. The vertices, edges, and faces of some polyhedron map onto the faces, edges, and vertices of its dual. Among the Platonic solids, the cube and octahedron are duals of each other, and the dodecahedron and icosahedron are also duals of each other. The tetrahedron is self-dual.
I will now derive what possible sorts of fair dice there are, a fair die being one whose faces act alike. Here are some names for such dice and similar polyhedra:
- Isohedral figure - face-transitive (faces look alike)
- Isotoxal figure - edge-transitive (edges look alike)
- Isogonal figure - vertex-transitive (vertices look alike)
The arguments that I will be using I will generalize to find plane tilings. Continuing with gaming metaphors, these are isohedral dartboards, fair ones with lookalike faces. List of convex uniform tilings has a list of them.
For a hyperbolic surface, one can also construct tilings, Uniform tilings in hyperbolic plane. These ones have the remaining parameter values, including faces with infinite numbers of vertices and vertices with an infinite number of faces. The hyperbolic plane is usually displayed as a disk, with most of the plane scrunched up near the edge of the disk. So one puts one end of an infinite-vertex face at the edge and one also puts an infinite-face vertex there.
There is an interesting interrelationship that some polyhedra have: duality. The vertices, edges, and faces of some polyhedron map onto the faces, edges, and vertices of its dual. Among the Platonic solids, the cube and octahedron are duals of each other, and the dodecahedron and icosahedron are also duals of each other. The tetrahedron is self-dual.