jack

The last post described why a d7 has to have one rotational symmetry, ie. be arranged circularly rather than spherically.

But the maths only applies on the surface of a *sphere*. On the surface of a *torus* it's different:

( Seven coloured torus courtesy of wikipedia )

Look. Seven regions. And look at the symmetries. There's one cycle that slides each block on to the one at the end, repeating after seven times. And there's one nearly perpendicular that slides a block onto a block above it, that repeats after three times. That's not a literal rotation, but it satisfies it morally.

But unfortunately, it's a property of the torus, not the map. You can make a d-anything the same way, without regard to the number. Draw a line that wraps the torus vertically (i) times and horizontally (j) times. (This one has (i,j)=(3,1).) That forms one (or more) strips that wind round the donut. Divide the strip into n equal parts. And lo, n-fold symmetry, together with i-fold symmetry.

Besides, it would be hard to *build* a torus die. Obvious throwing a donut doesn't land at a random point on its surface. I can think of two cheats, make a donut with a slidable surface (like a rubics cube, sort of), so the inner bit can come out to the outer bit. Then it's random. Or switch the central hole for the interior of the band to make another die, and roll both to get two coordinates, which define a random point on it. But you need a ruler or something, there's no way to mark them, let alone divide them into faces.

Well, one or other end of the horse, anyway :)

This probably won't be the end. But I finally stopped thinking about actual instantiations, and stepped back to consider what minimal theoretical conditions the d7 I want must satisfy.

To be theoretically fair, you need (a multiple of) seven orientations, each of which is equivalent to any of the others. Whether those are faces, or edges, and whatever other features there are, the fact that any must be equivalent to any of the others means they must be a symmetry group.

I think a "kind of spherical" die could be defined as one that has more than one axis of >2 rotational symmetry. Some messing around makes this seem unlikely (if you have two, the second must map to six more under rotation by the first, and vice versa, and those do not look like they can overlap) .

But fortunately, this is a completely solved problem, witness an exhaustive list from wikipedia: http://en.wikipedia.org/wiki/Point_groups_in_three_dimensions.

The only shapes with more than one genuine axis of symmetry correspond to the platonic solids. (Not just those, eg. a cube made up out of parallelagrams and other interesting dice qualify too.)

But there are no sevens to be seen. Thus, for 3d dice, without messing around with dynamics, the only ones with a multiple of seven can be around one axis, eg:

* The pencil, with ends flat, or curved to a point, or with seven-sided pyramid points.
* A seven-sided dipyramid, a la the octohedron for 4. (Seven edges round the middle, forming seven triangular faces with the points at the top, and the same at the bottom.)
* A twice-seven-sided Trapezohedron, a la d10. (Fourteen zig-zag sides round the middle, forming seven kite-faces with the point at the top, and the same at the bottom.)
* Gyroelongated_dipyramid, one of those padded out with more triangles round the middle. This probably looks most spherical, I'd like one of these.

NB: The actual 3d solution used is a pentagonal prism, which is nearly fair but not theoretically so.

1. No-one says rolling dice has to result in a *face*. It's obviously more convenient, but if you roll your dice in a V-shaped valley, you'll end up with them landing on an edge. And you could get a corner by using a conical well. (Though you need to shake it to make sure it doesn't get stuck.) Then any polyhedron with 7n equivalent faces, or edges, or vertices would do the trick. Unfortunately, I don't think it helps, I think you need 7N faces or regular septagons to get 7N edges or vertices, which you can't have.

2. You can have it in six dimensions, though. A 6-dimensional tetrahedron has seven 5-d-tetrahedral faces.

http://hjem.get2net.dk/Klaudius/Dice.htm
http://homepage.ntlworld.com/dice-play/DicePolyMath.htm

*sigh* Googling has shown more. *Of course* dice don't have to be platonic, the desire is isohedral -- where each face is symmetric with each other face but there's no requirement that the vertices. That satisfies the "obviously fair by geometry without any messy dynamics" requirement.

The good news is, there's lots of lists of lots of dice shapes. The bad news is, none of them introduce any new factors, the only numbers producible[1] are still just multiples of 2,3 and 5.

[1] Without cheating and rolling a pencil-like object with seven stable faces. That obviously *works* but is so clearly unaesthetic.

ETA: But that doesn't close the door entirely, though hope seems unlikely. For instance, consider a pencil with seven sides, and at each end, a point in a heptagonal pyramid. That's polyhedral, and fair amongst the seven sides, it's just not very spherical. We haven't yet shown that you can't make a spherical thing the same way, though it seems unlikely, seven isn't good at factoring.

http://www.maa.org/editorial/mathgames/mathgames_05_16_05.html
http://www.geocities.com/dicephysics/3sided.htm

We were sure you can generate a solid with seven faces, each of which takes up an equal "segment" from the centre (ie. if enclosed in a sphere about the centre of gravity, and projected out onto it, each face would take up the same area.). And so if given a random orientation and dropped, has an equal chance of landing on each face.

(In fact, there's subtleties, described in the link -- eg. the dice may roll from the bottom face, depending on its shape. In fact, he discovered, as the last post shows, some polyhedrons actually won't balance on some faces at all. But if you constrain your dice, I think you can eliminate those problems.)

However, we were worried that because the dice wasn't symmetrical (completely symmetrical dice have t be fair), what shape was fair would depend on the surface. The first guy also experimented with making d7s, and linked to the second guy, who actually experimented with non-polyhedral die and found it likely that the surface affected which face was favoured.

It's less obvious for a d7 because that's nearer to spherical (and might even have six identical faces) but looking at his calculations (calculating the chance of rolling from one face to another face) makes it seem unlikely the surface wouldn't make the distinguished face(s) more or less likely.

However, it's not clear if there might be some trick to work round it or not. The right question may yet to have been posed.

Entries tagged with dice

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