Convex Isohedra

Any die with 2n sides, with n>1, is possible. Most of such dice, however, are bipyramids and trapezohedra, which are not suitable for dice with more than 20 sides, since at that point they are essentially bicones, and are extremely difficult to read. Cylinders with small height (ie. coins) are often said to be two-sided dice, however they can, in rare circumstances, land on their side, and so don't really count. In some games, long n-gonal prisms are used since odd-numbered fair dice are impossible. This strategy has been used in games for thousands of years, and may actually predate dice. Nevertheless, since the ends count as sides, they obviously aren't included here. The most suitable dice are the platonic solids and catalan solids.

A Note on Circumsphere Volume

The circumsphere volume is expressed as a multiple of the volume of the polyhedron. The closer to 1, the closer the polyhedron is to a sphere, and the longer it will take for it to settle when rolled. Values under 2.5 tend to be very easy to pick up, and values above 4 tend to be somewhat difficult. This has to do with the amount of room the fingers have to move under the die. For cubes and tetrahedra, there is no room. However, cubes can be held at their sides by friction, whereas this is much more difficult with tetrahedra. Polyhedral dice such as d12s and d20s tend to be extremely easy to pick up, even for those with larger fingers, since there is much room under the die.

The value for octahedra is exactly pi. I believe the value closest to 1 would be that of the pentagonal hexecontahedron, but I have no information on the volumes and circumspheres of the polygons after the icosahedron. As for the ">2" value, I also can't seem to compute these values correctly (the solutions I am getting are definitely wrong), but the shape of the polygon as the number of sides approaches infinity is a bicone, and solving for that gives V=2. The actual values are unlikely to be much higher. With careful computation, I may be able to find the values for disdyakis dodecahedron and the disdyakis triacontahedron (since their central slice is just a polygon), but that work is for another day.


Sides Name Image Can Replace
4 Tetrahedron - ≈8.162
6 Cube ≈2.721
8 Octahedron d4 ≈3.142
10 Pentagonal
- >2
12 Dodecahedron d4, d6 ≈1.504
14 Heptagonal
- >2
16 Octagonal
d4, d8
18 Enneagonal
- -
20 Icosahedron d4, d10 ≈1.652
24 Pentagonal
d4, d6,
d8, d12
30 Rhombic
d6, d10
48 Disdyakis
d4, d6,
d8, d12,
60 Deltoidal
d4, d6,
d10, d12,
d20, d30
60 Pentakis
60 Pentagonal
120 Disdyakis
d4, d6,
d8, d10,
d12, d20,
d24, d30,
d40, d60