Patent for a five-sided dice [sic]

A patent for a "five sided dice" [sic].

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.