The Dice of Erich Zann

A friend and I recently spent a fair amount of time thinking about dice mechanics with a view toward game design, trying to create a roll with a probability distribution that was "nice" by various different kinds of standards -- the range of outcomes, the number of dice, the probabilities of extreme and median outcomes, the closeness of fit with a normal distribution, etc.

Eventually we decided that we'd like a distribution which approximated a bell curve, and had a 10% likelihood of rolling the mean and a 1% likelihood rolling an extreme result. A program was written to perform a search for suitable rolls of from 3 to 10 dice (3 being the minimum at which the roll is sufficiently bell-curvy, and 10 being too many dice), within an acceptable error tolerance on the probabilities.

The only result was 3d6. That's right: we'd spent all that time and effort only to come up with GURPS. That's when I gave up.

But then I saw this comic (click for source):

So there I was thinking "game design is futile and crazy", and suddenly there was a suggestion of a type of die-roll that actually IS crazy. Yes, thought I; let's have some non-Euclidean insanity dice!

Where to start? Well, how about four-dimensional dice? In 3D we have the five (and only five) regular polyhedra which, supplemented by the trapezohedral d10, comprise the set of normal RPG dice. In 4D there are six regular "polytopes", and they have 10, 24, 32, 96, 720, and 1200 sides. Okay, hyper-dimensional spaces are rather Lovecraftian in spirit, like the comic, but dice with those numbers of sides would still give flat -- i.e., "Euclidean" -- distributions. And besides, how do you simulate a 1200-sided die at the game table?

It turns out we can factorize those numbers of sides into products of existing dice; furthermore, it happens that if we require that each die is used at most once in any product, then the factorizations appear to be unique. Thus we obtain:

d10	= d10
d24	= d4 * d6
d32	= d4 * d8
d96	= d8 * d12
d720	= d6 * d10 * d12
d1200	= d6 * d10 * d20

Now let's have a look at the probability curves. The d10 has just the usual flat distribution, but check out the others:

d24
d32
d96
d720
d1200 <--- insane!

Now I just need to come up with a game where I can use these non-Euclidean hyperdice.