The patent was filed by Louis Zocchi, inventor of the Zocchihedron, which is a 100-sided die; the Wikipedia article indicates that the Zocchihedron isn't a fair die.

Will this one be?

Previously I wrote about how one might design asymmetric dice; Mark Dominus claims that "the probability that the hexahedron will land on face F is not proportional to the area of F, but rather to the solid angle subtended by F from the hexahedron's center of gravity." I'm not sure if I believe this. It seems reasonable, because it captures how the die is likely to rotate in the air, but dice bounce when they hit the table, and I'm not convinced that the "bouncing" behavior isn't chaotic.

Anyway, the patent indicates that the die is basically a triangular prism (although with beveled edges), with 1 and 5 on the triangular faces and the pairs (2,3), (2,4), (3,4) on the rectangular faces (thus 2, 3, or 4 will appear "upwards" when the die comes to rest); by symmetry, 1 and 5 should occur with the same frequency, as should 2, 3, and 4. So there is such a die.

Part of the patent reads as follows:

The present invention has been tested for fairness wherein different sizes of dice were included in the test ranging from 13-18 milimeters in thickness.... During initial testing, it was felt that the 14 millimeter thickness was the closest size to providing equally random outcomes for each of the five faces so that each face would occur one-fifth of the time. Specifically, 0,63 rolls were made of the 14 millimeter thickness test dice which yielded 6,152 rolls in which a rectangular silhouette was seen and 4,011 rolls which yielded a triangular silhouette. This means that the two triangular faces came up 4,011/10,163=0.3947 of the time. If the dice was perfectly fair, those faces should come up exactly 04000 of the time. Given the number of rolls, the uncertainty (one standard deviation) was estimated to be 0.0070 which indicates that the experiment detected no significant deviation from fairness.The actual standard deviation is more like √((10163)(.4)(.6)/10163 = 0.0049, meaning the results were a bit over one standard deviation from fairness; by the usual standards of statistics, though, it's still in a 95% confidence interval (i. e. within 1.96 standard deviations).

Eventually, it seems these will be manufactured at a thickness of 13.6 millimeters (which would prefer the triangular faces slightly more than the 14-millimeter thickness) but it is then stated that

It is believed that the dice may be ultimately manufactured in a range of size from 13 to 15 millimeters depending on the type of material they are to be used on.

It seems like a lot of trouble to have to have different dice for different purposes, which the inventor seems to think would be needed for fairness. (Perhaps this has something to do with the "bounciness".) There's a standard shape for a ten-sided die which could easily be used for this purpose (just label opposite sides with the same number), and from purely symmetrical grounds it's fair. I've been informed that rolling a ten-sided or twenty-sided (icosahedral) die and reducing mod 5 is standard among people who play role-playing games.

## 9 comments:

I think the existence of a unistable polyhedron http://mathworld.wolfram.com/UnistablePolyhedron.html is enough to confirm your "it's harder than that" suspicion.

Hi,

I came here from a link from Mark's site, and have much enjoyed reading everything you have to say.

I responded to Mark's post with a collection of links I'd found (mainly listed here

http://cartesiandaemon.livejournal.com/tag/dice ) on the subject of fair dice.

No five sided die is fair in the sense that a platonic solid or isohedral solid is, in that the chance of each face (or equivalent set of faces) depends on the surface it's rolled on.

There are 7-sided dice produced in the same way, as a pentagonal prism.

These are generally considered "good enough" but not perfectly fair.

I don't know how much the error is, or how much it might be reduced by having different dice for different tables.

I realize that I'm pulling an old post out of the archives of yesteryear here, but I just wanted to ask, why not an elongated pentagonal dipyramid? It is essentially a pentagonal prism with pentagonal pyramids on the ends. Due to its radial symmetry and the impossibility of resting on the pyramid faces, it seems it should be a completely fair D5.

(Forgive the double post, please. I was unable to edit out the awful typo in the previous version).

I bought one of those, and if you roll it a bunch of times on a flat hard surface, it's pretty consistent. If I roll it 5 times, I usually get two triangle sides (1 or 5) and three of the others (3, 4, or 5). I think the trick is in the angle in which the edges were cut. The way to roll it is to kind of throw it a little. like don't let it slide out of your hand onto the table, but give a little toss from some inches above.

I also got a 7 sided die, which is a lot less consistent, usually because it often rolls like a cylinder.

Good day !.

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