Wikipedia:Reference desk/Archives/Mathematics/2020 December 8

Welcome to the Wikipedia Mathematics Reference Desk Archives
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages.

December 8

Could the Catalan Solids be used to create a fair die?

I know that there are usually 5 types of dice used in board games (the platonic solids), but since the faces of the Catalan solids are the same shape, could you be able to make a die with it, having an equal probability to land on each side? Or do the faces have to be regular? Keep in mind that I'm not a mathematics expert, so keep the answers not too complicated. Thank you. — Preceding unsigned comment added by Eridian314 (talk • contribs) 18:26, 8 December 2020 (UTC)

Well, it's not really a math question per se. It's a (extremely complicated, if you really take everything into account) physics problem. I would guess that you could tune things to get very very close to fair, by adjusting weights and such, but probably not perfect.

For that matter, even the standard cubical "fair die" is unlikely to be perfectly fair in practice. The spots affect things somewhat, as do tolerance errors in manufacture. Really "fair die" is just a way of saying "I'm setting all my prior probabilities to be equal". It's an ideal simplification to let you work the problem, not an actual physical thing. --Trovatore (talk) 18:42, 8 December 2020 (UTC)

Yes; this is mentioned in the first paragraph of Isohedral figure. --116.86.4.41 (talk) 19:33, 8 December 2020 (UTC)

With the standard six-face die, whose idealized form is a homogeneous solid cube, the value after a roll is that of the face that comes up. For dice based on Catalan solids, or convex isohedral solids in general, you may have the problem that once the die comes to rest there is no face that is "up". This is very obvious for a tetrahedral die. If the faces have no parallel opposite faces, the only symmetry-breaking choice is to read the bottom face. In the real world, this is a perfect opportunity for sleight of hand. In an ideal Newtonian universe with uniform gravity, no air resistance and deterministic physical laws, the randomness of an ideal (but not perfectly elastic) die is supplied by the randomness of the initial conditions when the die is cast. The initial conditions can be given as position, velocity, spatial orientation and spin, all together a 12-dimensional parameter space. (For solids in general, three can be eliminated by symmetry considerations: for position, only the height above the surface plane is relevant, and the initial setup can be rotated around a vertical axis.) The die also has intrinsic physical parameters; let us give it a diameter of one unit and a mass of one unit in a gravity field whose strength is one unit. Given the initial conditions the die has a certain energy (potential plus kinetic), of which it will loose a fraction on each bounce. Let us also fix the initial energy and the elasticity parameter. Then, given a uniform probability distribution over this parameter space (meaning it is invariant under the geometric symmetries of the Newtonian universe that leave the surface plane in place and do not swap "up" and "down"), the face transitivity of an isohedral die guarantees that all outcomes are equally likely. Persi Diaconis has experimentally demonstrated the limited validity of such idealized assumptions.  --Lambiam 08:23, 9 December 2020 (UTC)

There are solutions to the problem of not having a top face, including labeling the vertices instead (Commons:File:D4_Numbers_at_top.png) or placing face labels around the face (Commons:File:4-sided_die.jpg).

Labelling the vertices works for the tetrahedron because there is a one-to-one correspondence between faces and vertices. No other polyhedron has one. Reading labels around the bottom face of pentagonal hexecontahedron is no mean feat.  --Lambiam 15:48, 9 December 2020 (UTC)

Among the Catalan solids, only one lacks opposite faces: the triakis tetrahedron. In its case, what ends up on top is a pair of sharp corners of two adjacent faces, so the label can be placed on one or both of those two corners. --116.86.4.41 (talk) 01:58, 11 December 2020 (UTC)

There are polyhedra that are monohedral (all faces are congruent), but not isohedral. Examples of this are the duals of the pseudo-uniform polyhedra, such as the pseudo-deltoidal icositetrahedron. The above argument of fairness under idealized conditions then does not go through.  --Lambiam 09:30, 9 December 2020 (UTC)

It looks like the (monohedral) snub disphenoid will snub fairness. Even more obviously unfair: the gyroelongated square bipyramid. (I have not done the maths that would prove it unfair.) A recent paper by Wikipedist David Eppstein shows twisted gyroelongated monohedral constructions that also seem unfair to the honest gambler.  --Lambiam 09:58, 9 December 2020 (UTC)

Isohedral (face-symmetric) dice are fair. Monohedral dice, not necessarily. Yes, the shapes in my paper are monohedral but not necessarily isohedral. The Dice Lab (with whom I have no connection other than admiring their work) have taken advantage of the existence of isohedral shapes other than the Platonic solids to find interesting ways of distorting dice shapes while keeping them fair. —David Eppstein (talk) 18:17, 9 December 2020 (UTC)

More generally, a fair die is the intersection of a half-space (containing the origin!) and its images under some point group. —Tamfang (talk) 02:08, 12 December 2020 (UTC)

Does this characterization include all fair dice, such as the non-symmetric dice of section 3 of the "Fair Dice" paper^[1] of Diaconis and Keller?  --Lambiam 04:15, 12 December 2020 (UTC)

The problem with the analysis in that paper is that it uses the intermediate value theorem to prove the existence of some shape so that the die is fair, but the IVT does not tell you what that value is. With non-symmetric dice the actual probabilities depends on the physical properties of the dice and the surface they are being rolled on, and presumably other factors such as velocity and angle the dice are traveling at when they hit the surface. The complexity of such an analysis would make it nearly impossible to carry out theoretically, so (as the paper mentions) it would be done by experiment. Symmetry allows you to ignore the physical properties because they affect the probabilities symmetrically. But there are other construction using the IVT. For example look at all square pyramids with a give volume parameterized by the ratio between the side of the square and the height. At one extreme the die will land on the base with a probability of .5 and at the other with a probability of 0, so there is some ratio in between where the probably is .2. But, again, finding that value is a problem in physics, not mathematics. Other such construction are possible as well, and I doubt enumerating them has much value. --RDBury (talk) 21:18, 12 December 2020 (UTC)

Yes, that's kind of the point I was getting at in my first response. It's true that I missed that there's a formal sense in which the Catalan solids have no distinguished face, if for any two faces there's a rigid motion that takes one to the other and makes the image match up setwise, and that's probably what the question was actually about, so my bad on that point.

Even for the symmetric ones, it's still a sort of stylized answer, of course, where you take some things into account and ignore others. There's not really an obvious mathematical notion of what it means to roll a die, so it's not clear that the symmetry really means that all outcomes are equiprobable. Are all the initial states equiprobable? --Trovatore (talk) 22:19, 12 December 2020 (UTC)

Yes, and many have been! Double sharp (talk) 12:58, 15 December 2020 (UTC)