# Rules for polyhedrons

This is probably something that this biologist does not remember from high school. So shoot me. I’ve thought about it a little and not come up with much. What polyhedrons can be made with all sides being equivalent polygons? Ok, you’ve got a three-sided pyramid (with the bottom the fourth side) and a cube, but I can’t think of any way to make a five-sided polyhedron with all sides the same, and I don’t imagine it is possible. What about 9 sides? Could I make a game with a nine-sided die with all sides the same?

I mean polyhedra.

The five regular polyhedra, as outlined by Euclid (I’m working from memory here) are:

Tetrahedron: 4 triangular sides
Cube: 6 square sides
Octahedron: 8 triangular sides
Dodecahedron: 12 pentagonal sides
Icosahedron: 20 triangular sides

IIRC, these are the only solid figures that can be made with sides whose angles and edges are equal. For other solids, see the work of R. Buckminster Fuller

Someone will be along shortly with links to illustrations of these figures

There are actually 9 regular polyhedra. Those five are the Platonic solids, which are convex figues. There are additionally 4 concave solids, pictured here in the top row.

If you require that all faces be the same and that they all be regular polygons (i.e. equilateral triangle, square, or pentagon) then there are five Platonic solids: the tetrahedron, the cube, the octahedron (formed from 8 triangles), the dodecahedron (12 pentagons), and the icosahedron (20 triangles). If you extend this to allow different types of faces meeting at a vertex, you get the 13 Archimedean solids, but these wouldn’t be good dice because their faces are different shapes and would probably come up with different probabilities. It’s possible to relax the restriction that the faces be regular polygons as well; if we require that all the faces be rhombi, for instance, we get the rhombic dodecahedron and the rhombic tricontahedron. These would be useful as dice, and indeed I’ve seen the rhombic tricontahedron used as a 30-sider.

I don’t know if this is an exhaustive list of “polyhedral dice”, though. Hopefully someone else who knows will be along shortly.

On preview: hey, commasense was right, I do have figures.

Great Links! You guys are great. I probably could have come up with some of these in my own short search if I had your command of the lingo. Use the wrong words in search and get no results! go figure. Thanks guys.

And let’s not forget Euler’s formula for the 5 Platonic polyhedra:

Vertices - Edges + Faces = 2

Here’s a good link for the formula and the 5 polyhedra:
http://www.math.ohio-state.edu/~fiedorow/math655/Euler.html

They include the “Buckyball” for a 6th polyhedra. It might follow Euler’s formula but the “Buckyball” does not fit the definition of a Platonic Polyhedron.

According to http://en.wikipedia.org/wiki/Platonic_solid. There are only the 5 platonic solids which are convex and have the same number of the same polygon meeting at each vertex but you get other shapes if you allow vertices with 3 OR 4 triangles in the same polyhedron, or concave shapes, etc.

Whether you allow these depends on why you’re wondering. I think, but don’t know, that a platonic solid is what you want for a good dice. (Obviously other shapes could work, but it would be hell proving that.) There are no other p.s. As for the other less regular shapes, well I don’t think there can be shape with 5 side all the same, but I’m not sure if you could do something for 9.

Of course, if you want a perfect 5 sided dice you could just use a regular isocahedron and label 1 to 5 four times.

BTW - I’ve heard both “polyhedra” and “polyhedrons.” I don’t know about anyone else, but I wouldn’t correct either.

Hijack: the formula actually works for almost any shape (that doesn’t have a hole all the way through).

AFAICS they don’t say those are the platonic solids, they just demonstrate the formula on them.

It works for any polyhedron, even irregular ones and even ones with holes if you change the right hand side to 2-2Genus (number of holes).

For example, there are 8 “deltahedra”, which are polyhedra all of whose faces are equilateral triangles, but where the number of faces meeting at each vertex isn’t necessarily constant. Three of the Platonic solids are deltahedra (the tetrahedron, the octahedron, and the icosahedron, with 4, 8, and 20 faces respectively) but there are also deltahedra with 6, 10, 12, 14, and 16 faces. Two of those would definitely be usable as dice: the 6-sided deltahedron (which is two tetrahedra glued together) and the 10-sided deltahedron (which is two five-sided pyramids glued together along their bases). In fact the 10-sided deltahedron is similar (but not quite the same as) the 10-sided die that is actually used in some role-playing games.

It is easy to make perfect dice of any number of sides. First consider an even number, say 2n. Start with n congruent isoceles triangles. They can be arrayed in a pyramid. Duplicate the pyramid and paste them together, base to base. This gives you a die with 2n equally likely faces. By numbering opposite faces the same, you can instead have n equally likely possibilities.