A die, usually, is a common word for a hexahedron. In recent time, some games use other polyhedrons than a hexahedron as random number generator, and these other polyhedrons are also called dice. But not every polyhedron can be used as a random number generator, or at least not as a laplacian random number gernerator (that means, with all sides having the same probability), that means, as a die. So we may define a die as a polyhedron with certain symmetrical properties to ensure it to be laplacian.

Definition: A die is a polyhedron with the property that if A and B are two areas of the polyhedron, than there is a translation, combined with a turning, and occasionaly combined with a mirror fliping that maps the polyhedron to itself (i. e. is a symmetrical mapping) and maps A to B.

In the following, we will regard two dice as the same if there edges form the same graph. We will say that they are two different polyhedrons, but the same die. Note that two different polyhedrons forming the same die can differ dramatically in there properties, for example, one of them can be konvex while the other isn't. But we will see that there is always one represantation of a die that is in fact konvex. Also note that every die is topologically equivalent to the sphere (so being the same die is stronger than being topologically equivalent, since all dice are topologically equivalent). The picture below shows an example of a die, once in its konvex version, one in its non-konvex version, and the usuall hexahedron die together with the graph of its edges.

In the following, I will introduce a method of indexing dice. A die consists of areas, every area consists of edges and vertices, and the vertices are joined by the edges of different areas. So a vertice can be characteristed by the number of edges it conects. For example, in an ordinary hexahedron, every area has 4 vertices, and every vertice conects 3 edges. Now 2 of these three edges come from the area itself, only one edge is distributed by other areas. So we will give every vertice the value 1, and the whole area will get the code (1, 1, 1, 1). Now we consider the area of an octahedron: every vertice conects 2 edges besides the two edges distributed by the area itself, so the area will get the code (2, 2, 2). Now consider a polyhedron consisting of two tetrahedrons glued together: Every area has one vertice with the value 1, and two vertices with the value 2. So the code is (1, 2, 2) (or (2, 2, 1) or (2, 1, 2), that doesn't matter). See picture below:

Now every die can be characterised by its code. The other way round, we can use this code to index all possible dice. For all codes, there are four possible cases:

1. The code leads to the graph of a polyhedron topologically aquivalent to the sphere. In that case, we have discovered a die.
2. The code leads to an invinite graph topologically aquivalent to an euclidian plane. We may call such a plane a "die plane".
3. The code leads to an invinite graph topologically aquivalent to a hyperbolic plane.
4. The code does not lead to any graph at all, it is, for some reason, an "impossible code".

There are some rules for dice codes: as you can see in the picture, any code of the form (1, a, 1, b) is impossible unless a=b (or a or b=1, of course). More generally spoken, it can be proven that any code of the form (2k+1, a, 2k+1, b) is impossible unless a=b. Another rule you can easily verify is that any code of the form (1, a, b, b) is impossible unless a=b (or b=1, of course). Another rule is that (1, 1, a, b) is impossible, regardless wether a=b or not, if they are both larger than 1, as you can check out easily. With these rules, we can exclude certain codes as impossible, and we will discuss only the remaining legal codes.

Another rule deals with hyperbolic planes. If (a1, a2, ... aJ, ... aN) is a code of an euclidian or hyperbolic plane, than (a1, a2, ... aJ + 1, ... aN) will be in any case a hyperbolic plane. Also, any area with seven vertices or more leads to a hyperbolic plane, so if we are not interested in hyperbolic planes, we can stuck to codes with six vertices or less.
So we will have, with some luck, a limited field of investigation: we start with the codes (1, 1, 1), (1, 1, 1, 1), (1, 1, 1, 1, 1) and (1, 1, 1, 1, 1, 1), increase the values, check wether we received a sphere, an euclidian plane or a hyperbolic plane, and stop if we start getting a hyperbolic plane. That way, we should get the codes of all possible dice and dice planes.

Doing this, I received the following results: There are two infinit families of dice, seventeen single cases of dice, and ten dice planes.

Some words about dice planes: they differ slightly from usual parquet floorings. For example, a parquet flooring with rectangles in a brick sample is indistinguishible from a parquet flooring with sexangles (nice word, "sexangles"):

So there are more different parquet floorings than dice planes.

And now the results:

The first infinite familiy contains dice of the form (2, 2, n), with arbitraire n. An example of this familiy is the octahedron, that is, (2, 2, 2), or the two tetrahedrons glued together (2, 2, 1) we mentioned above. The members of this families are always two pyramids glued together. Number of areas: 2n+4; number of edges: 3n+6; number of vertices: n+4.
The other infinite familycontains dice of the form (1, 1, 1, n), with arbitraire number n. A famous example of this family is (1, 1, 1, 1), the hexahedron or ordinairy die. The members of this family consists of two pyramid-like things, glued together in a somewhat strange way. Number of areas: 2n+4; number of edges: 4n+8; number of vertices: 2n+6.
The seventeen remaining single cases:

code	areas	edges	vertices
1,1,1	4	6	4
3,3,3	20	30	12
1,4,4	12	18	8
2,4,4	24	36	14
3,4,4	60	90	32
2,4,6	48	72	26
1,6,6	24	36	14
2,6,6	96	144	50
2,4,8	120	180	62
1,8,8	60	90	32
1,2,1,2	12	24	14
1,2,2,2	24	48	26
1,2,3,2	60	120	62
1,3,1,3	30	60	32
1,1,1,1,1	12	30	20
1,1,1,1,2	24	60	38
1,1,1,1,3	60	150	92

Every even number of areas greater or equal 4 can be realised (if you want a die with an odd number of faces, say n, you have to construct a die with 2n faces and number it from 1 to n two times). The die with four areas, the tetrahedron, is a somewhat special case: it is the only die in which you can construct a line going through the middle of the polyhedron and the middle of one area without this line going through the middle of another opposite area (as with all other dice), but through a vertix. Now for sake of completness the dice planes:

4,4,4*
2,4,10*
1,10,10

2,2,2,2*
1,4,1,4
1,2,4,2

1,1,1,2,2,
1,1,2,1,2,
1,1,1,1,4

1,1,1,1,1,1

The three codes marked with a * can be colored with only two colors, the other seven codes can be colored with a minimum of three colors.

A platonic polyhedron is a special case of a die: a platonic polyhedron is a die with a code with all numbers equal, like (a, a, .... a). The five platonic polyhedrons are (1, 1, 1), (2, 2, 2), (3, 3, 3), (1, 1, 1, 1) and (1, 1, 1, 1, 1) (tetrahedron, octahedron, icosahedron, hexahedron, dodecahedron). So this is another proof that there are only five platonic polyhedrons [Strictly spoken, it is not really another proof, since it uses basicly the same argument as Euklid]. There are also three objects we might call "platonic planes", that is (4, 4, 4), (2, 2, 2, 2) and (1, 1, 1, 1, 1, 1) (parquet floorings with regular triangles, squares and sexangles).

A last remark: there is another way of constructing dice for game play: one could construct a polyhedron consisting of two different kinds of areas, m*n areas of the one kind and k*n areas of the other kind (with 1 as the greatest common divisor of m and k). The polyhedron should be constructed in a manner that every area of the one kind is as probable as any other area of the same kind. Then the areas of the first kind are numbered from 1 to n m times, while the areas of the second kind are numbered from 1 to n k times. This could serve as a die with n different areas.

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