Imagine a five-sided die in the shape of a pyramid with height H, with four triangular sides and one square base with sides of length B. Further imagine that this pyramid is very tall and narrow, with H/B approaching infinity. Now you cast the die with sufficient height and spin to assure randomness. Based on the general shape of something long and skinny, we can intuit that the probability of landing upright on its square base is very nearly zero, and the probability of landing with any one of triangular sides facing down is about 25%.
Now change the aspect ratio, so that H/B approaches zero. Cast the die again. Based on the general shape of something very flat and almost coin-like, and we can intuit that the probability of landing upright on its square base is about 50%, and the probability of landing with any one of the triangular sides facing down is about 12.5%.
Somewhere in between these two extremes of H/B there’s a numerical value for which we have a perfectly fair five-sided die, i.e. one with a 20% chance of landing with any of its five sides facing down. Is there an analytical method to identify this value of H/B, or would we have to run physical trials (or computer simulations) over a range of H/B values to identify the H/B that produces a fair die?
There is an old thread about this, with some ideas, but nothing really definitive. There is also a suggestion there to try rolling plastic models to check, and if you are handy with 3d modelling you could do a computer simulation.
Kevin Cook has the worlds largest dice collection which I think is on the computer as DiceCollector.com. He told me that a Doctor of Mathematics who taught at a college in Canada, had built a dice testing machine to see if dime store dice rolled as randomly as casino dice…
Because I didn’t know how thick to make my D-5, I sent him 11 prototypes, each of which was 1mm thicker than the next. After several months had passed, he told me I needed to make my die 13.85mm thick in order to assure that it rolled every face an equal number of times.
I phoned him for more details. Obviously 1 mm is a very small amount, and he wanted me to take 1/15th of a millimeter off of the 14mm thick prototype. “How did you come up with such a strange finding?”, I asked. He said, after his machine had rolled the 10mm thick die more than 5,000 times, he plotted its results. Then he rolled the 11mm thick die more than 5,000 times and plotted its result. Then the 12, 13 and 14mm dice were rolled. He repeated this testing on 6mm thick plastic and 12 mm thick plastic to confirm the performance results, which indicated that a die that was 13.85mm thick would roll each of its faces an equal number of times…
If you make the base of the die conical (so it cannot land flat on that side at all) and the conical base shorter in height than the pyramid made by the five flat sides, it cannot rest on the conical base either, but must tip onto one of the flat sides.
The only remaining problem being that there will be no upward face due to the odd numbered sides, so you probably have to number the edges
It would be interesting to determine if trial and error found any regularity in the progression of n-sided dies. The proper height to perimeter ration for four, five, six…sided dies made as “pyramids”.
Although not in the spirit of the OP: Years ago I saw a game that came as a cardboard cutout. The “die” was a cardboard hexagon with the numbers 1-6 printed along the edges - and the instructions were to stick a pin through the center, spin it like a top, and let it fall over on one side. So a five-sided “die” could be made the same way.
Does anyone make a ten-sided die - of two five-sided “pyramids” back to back with pentagons for bases?
It’s probably possible to calculate the necessary dimensions so that each side subtends the same spherical angle. I might take a crack at it later today.
I don’t think, however, that this would guarantee that it would be a fair die if rolled. Once it lands on the table, it starts to lose energy due to friction and deformation of the table. It will continue to roll over from face to face until it doesn’t have enough kinetic energy to do it again. But if the center of gravity is closer to the square face than to the triangular faces, then the die would need more energy to go from a triangular face to the square face than vice versa. This would (I suspect) create a bias towards the square face.
If you rolled the die on a surface that was very “squishy”, so that almost all of the kinetic energy was dissipated the first time the die touched the surface, then you could probably get away with just looking at the solid angles subtended by the surfaces. But for a more normal rolling surface, the distribution of outcomes is going to depend somewhat on the solid angles and somewhat on the energy dissipation; and which fact wins out will be situation-dependent.
I wouldn’t trust any physical die without constructing and testing it. @Steve_MB 's link was not about a pyramid, though. In my link, one of the posts suggested a construction where the solid angle subtended by the base from the centre of mass is one-fifth of a sphere (the idea, I suppose, is that it will then have an equal chance of landing on that base and hopefully not bounce around too much). That might be a starting point; back of the envelope gives me H/B approximately 1.675 but I probably forgot to multiply by 2 or flipped a sign somewhere so you had better check from scratch.
They might, but conventionally D10s are pentagonal trapezohedrons, not just two pyramids. The only variant D10s I’ve seen are prismatic ones and the crappy spherical ones, never seen just pyramidal ones. And there’s no need for quotes around pyramid, a pyramid can have any polygon as a base.
Yes, I was just talking about the iterative methodology.
I know the OP’s intent behind the question is geometrical rather than stochastic, but if it were stochastic (i.e., the desire to have, for gaming purposes, a die that can choose among five different outcomes with equal probabilities), then there’d be a simpler solution: Just take a regular icosahedron and label its twenty sides such that each of the five outcomes shows up on four sides.
Now how did he conclude that? The final prismatic die he tested came up with a \chi^2 statistic of 0.275 for 4 degrees of freedom, and the long die wasn’t tested at all.
Sure, you can do that too. The only disadvantage I would see compared to the twenty-sided dice (common in fantasy role-playing games, so they’re easily available) is that one out of six rolls would have to be discarded and repeated, so there’s some waste involved: You’ll need to have 20 % more rolls of the die to get the same number of valid results.
However, a long die must be fair if it’s manufactured accurately enough. The cross section is a regular pentagon and there’s no reason why any side would be favoured.
In practice, accuracy of manufacture will affect the fairness of any dice.
This is a bit like the fair 3 sided coin (2 faces and an edge with a desired outcome of 1/3 on each of those 3 possibilties) that I think Matt Parker investigated. I think the issue probably generalises to: if the form is not regular and the faces not identical, there is no way for it to be completely fair and the unfairness will manifest in how high it is thrown, how much spin etc
