Ultimately, it comes down to this: if you remove the numbers, it should be possible to rotate the die in at least N different ways such that the die appears identical. It is even better if the number is higher than the face count (as with an ordinary 6-sided die, which has 24 identical orientations), but an equal number is sufficient.
I’m wondering now how mirror symmetry fits into this. One could design a die with half the faces being a mirror image of the other. That might be a fair die on a typical surface but unfair on a surface that itself lacks mirror symmetry.
Yeah, for example you could take a pentagonal prism and squash it - the faces might be identical, but it would no longer be fair as it’s more likely to end up landing flat than tall (that is, with the centre of mass lower rather than higher)
I think the edges are being underconsidered as well. The way an edge catches on a surface and transfers momentum is going to have a large impact on the scrambling effect. And this will manifest differently on a smooth wood surface vs. felt, etc. The only way to be sure that all the effects apply equally is for it to have perfect rotational symmetry. The Platonic solids do one better by having extra symmetries on top of the minimum.
I have, in fact, made a bipyramid d10. It doesn’t end up with a face on top, so I put the numbers on the edges instead. I’ve also designed, but never yet actually made physically, a few others that work equivalently.
Mine are the only bipyramid d10s I’ve ever seen, but The Dice Lab has bipyramid d6s for sale, which is the same idea. I highly recommend them, by the way: They’re the only company I’ve found that takes dice as seriously as I do.
And I’ll agree with others that you can’t really make a properly fair die unless it’s an isohedron (which, incidentally, all of the Dice Lab products are, even if it’s not initially obvious). Given a surface to roll on, the Intermediate Value Theorem guarantees that there must be some fair proportion… but there’s no guarantee that the same proportion will be fair when rolled on a different surface. A die that’s fair when rolled on a marble slab might not be fair when rolled on a wooden tabletop, or a springy mousepad, or the cardboard cover of a book, or whatever other surface you might be rolling on.
I’m unconvinced that isohedrality is a sufficient condition for fairness . A rolling surface which is itself not mirror symmetric could have different behavior for half of the faces.
Not quite true; they have this disclaimer on some of their products (the d18, d22, d26, and others):
Note that this design is not isohedral.
I’m guessing that all products without the disclaimer are isohedral, though.
ETA: Never mind. It appears that “Dice Lab” dice are a specific subcategory of their shop, which are all isohedral. Other subsections don’t make the same guarantee.
I think of them as dodecahedrons with the pentagonal faces stretched toward a point. In fact, the ten-sided dice I’ve seen never come to a point, but truncate that long spindle to a tiny pentagonal face too small for a thrown die to land on.
If you want truly equal probabilities for all five numbers, by symmetry alone a doubly-number 10-sided die like this or a 5-sided “pencil” (with pentagonal cross section), as shown in one of the above posts, are clearly your best choices.
Or you could just use an ordinary 6-sided die and simply ignore and re-roll if it comes up a “6”, but that’s probably too tame to consider.
Strictly speaking, you’re probably right: Given a die with faces that are congruent scalene triangles, for instance, one could make a surface composed of indentations in the shape of one of those triangles, tesselated across the surface, which would favor the faces that fit into those indentations. Realistically, though, it’s difficult to imagine asymmetric rolling surfaces being particularly relevant.
Sure, but I think of fairness as being a kind of game: one party is trying to design a die with outcomes that cannot predicted at better than chance, while the other party is trying to design a surface and rolling technique that subverts that (within certain limits when it comes to rolling). A truly rotation and mirror symmetric die, weighted perfectly, is perfectly fair. Other dice are less so.
I considered the surface with face-shaped imprints, but I think even something like a felt surface that is matted in a particular way could influence the roll, say by making transitions on the long edge easier in one direction.
Eh, even a “perfectly fair” die can be subverted with the proper “rolling technique”. Like, say, holding the die in the preferred orientation and dropping it straight down. Or for a less extreme example, the first d12 I owned had the 7,8,9,10, and 11 arranged around the 12, and I found that I was able, with a bit of practice, to consistently get one of those six faces, and thus guarantee an above-average result. This problem can be mitigated (though not eliminated entirely) by proper arrangement of the numbers on the faces.
As I said, “within certain limits.” Perhaps player 2 can roll the die however they wish, but player 1 gets to choose the initial orientation (and player 2 has to commit to a type of roll beforehand). Clearly, one can always have a “degenerate roll” that consists solely of the player setting the die down on the preferred face.
None of this is really a problem in practice of course; I just like to think about how one would achieve true fairness. No one uses fair pie-cutting techniques, either.
Part of the condition in my OP was that the die be cast with sufficient height and rotation to assure randomness, at least as regards any influence from the person casting the die.
A square pyramid with the required proportions was 3-D printed out of plastic. It was noticeably lopsided. Note: next time, spend a few seconds filing it down, and/or figure out how to 3-d print a polyhedron so that it does not look like a cross between an overheated pyrometric cone and Crazy String.
Die was rolled on to a desk a bunch of times.
Results:
1
2
3
4
5
31
32
15
28
22
(base of pyramid = 5)
So even this piece of junk is, in limited testing, not biased with 99% significance. Testing was aborted after it was noticed that side “3” was not stable (also the data seems to reflect that)
Die still looks amateurish, though a token effort was made to smooth it off.
1
2
3
4
5
45
47
47
50
48
I am willing to declare that it is plausible that an appropriately manufactured pyramidal die may be good enough for Dungeons & Dragons or similar, even though it is by definition not completely symmetrical, so you would not want to use it in any casino games.
@DPRK, thank you for going above and beyond in analyzing this pressing issue.
Would you consider posting pics of your die? I’m having a bit of trouble visualizing it. Plus, as a long time table top gamer, I like looking at quirky dice.
This was supposed to be a solid pyramid with H/B = 1.6749, but, again, first of all I am not positive that is the right value (it certainly isn’t if you take into account the bevels or whatever you would use in a real one, but maybe a pro CAD program could automatically figure out the centre of mass and solid angles and adjust the value), and, secondly, my 3-d print was shit (but that’s OK as this was only meant to be a simple test of feasibility). I did not attempt a proper 3-d design (just exported a polyhedron) or to use a resin printer or anything like that.
. A rolling surface which is itself not mirror symmetric could have different behavior for half of the faces.