There really IS a polyhedral d1

[jack]

In retrospect, it's obvious. That is the best d1 (one-sided die).

ETA: We went to a little difficulty defining a d1, but it was unspokenly assumed you couldn't build one without cheating a little, eg. ingeniously using a sphere... But this works.

It satisfies all the requirements for a dice. It's solid, uniform, all the faces and edges are straight. It's equally fair rolled on any surface provided with random initial orientations. And if you paint a "1" on the only stable side, it always stops on it!

That's better than we managed for d7! :)

Comments

[ravingglory.livejournal.com]

one die, two dice. Normaly I try to avoid pedantry, but being pedantic at you is hard to resist.

[cartesiandaemon.livejournal.com]

And I like giving you the opportunity :) (Seriously, I enjoy knowing that, but whenever "a die" looks like it might be ambiguous I just use "dice" and damn the tradition :))

[pseudomonas.livejournal.com]

When is it ambiguous? when you might mean a device for shaping metal things?

[cartesiandaemon.livejournal.com]

I was thinking of the verb, actually. It's not literally ambiguous, I just didn't like the way it sounded.

[cartesiandaemon.livejournal.com]

How's the field trip?

[simont]

No, that's not better than we managed for d7: it's exactly as good as the best known d7, i.e. a heptagonal prism with stuff stuck on the ends to make them unstable. Both apply precisely the same dodge, namely having n stable faces plus a bunch of unstable ones.

[cartesiandaemon.livejournal.com]

You're right, sorry, I was carried away with the coolness of the idea, and didn't really stop to consider it.

OK, how about http://blog.sciencenews.org/mathtrek/2007/04/cant_knock_it_down.html ? If they're right, they've a requirement that their shape has at most one stable and one unstable balance point (which it must have, at least). Which seems a good formalisation of the "prisms are cheating" criterion. And have found a shape, but say its unknown if there can be a polyhedron that does it.

[simont]

Ooh, and hang on: that approach also produces a new approach to a d0. A d0 merely has to have no stable faces, only unstable ones; so you can throw it, but it never lands. For example, the traditional cat with buttered toast strapped to its back :-)

[pseudomonas.livejournal.com]

Or never stabilises - consider a hamster-ball with an immortal and unsleeping occupant.

[simont]

Hmm. If you change the rules only very slightly, one could argue that a perfect sphere is in fact a d0, on the grounds that it can't be said to have definitely landed in any particular orientation until it's at a stable equilibrium – and a sphere is always in neutral equilibrium, so by that definition it always lands cocked.

[pseudomonas.livejournal.com]

I was assuming that we'd ruled out spheres and was picturing a d100 with internal rodent.

[simont]

AIR we previously considered spheres but assumed that they had one "face" comprising their entire surface, which was always facing upwards when they landed. I'm now revising that definition in the light of clearer thought about what it means for a die to land in a particular way.

[pseudomonas.livejournal.com]

They just feel like the wrong kind of stuff. Similarly, making a cube full of helium that would not land if thrown in air would be cheating.

[cartesiandaemon.livejournal.com]

I was just in the process of suggesting that. Is it cheating more than buttered cats? :) I imagined it perfectly weighted, so it floats up, and bounces off the ceiling, and a few walls, and you *think* it's rolling, and you think it's going to stop, but it stops just sort of floating.

But I want dice that can be any density.

In fact, that could work. Forget helium, throw a solid polyhedron in a closed room with no air and no gravity. It'll roll around, and eventually all the kinetic energy will be absorbed, right? At which point it will be flat to a surface? But if that's not the the floor, you can't read it! :)

[pseudomonas.livejournal.com]

I don't want to play RPGs in a room with no air!

[cartesiandaemon.livejournal.com]

ROFL. That is a perfect sentence :)

[pseudomonas.livejournal.com]

It gave me a chance to use the rodent icon, too.

[cartesiandaemon.livejournal.com]

I thought so. It's sweet :)

[cartesiandaemon.livejournal.com]

I think we *did* argue that :)

[cartesiandaemon.livejournal.com]

Yes, I was wondering something like that.

It doesn't even matter if it lands, just so long as its not stable on any face -- but I'm fairly sure you can't construct a geometrically astable polyhedron. A sphere or frictionless-elastic-collision-thing is the nearest.

the traditional cat with buttered toast strapped to its back :-)

That works.

DM: OK, roll for sanity failure. Fetch your favourite dice and-- Thomas, what's that?
Thomas: My d0.
DM: Your d0?
Thomas: I don't want to roll a sanity failure.
DM: Well, ok. On your head be it.
Thomas: OK. Ow! Ow! Ow!
D0: Yowl!
Thomas: Agh!
D0: Yowl! Miaow! Yowl! Hiss!
Thomas: Ready...?
D0: Splat.

[simont]

Indeed, you can't construct a convex polyhedron with no stable face: consider expanding a sphere gradually out from the centre of mass until it touches a face for the first time. It must be tangent to that face, and hence if the solid rests on that face then the CoG is directly above the point of tangency and the face is stable.

If the polyhedron isn't convex, then that proof doesn't work because the first face encountered by the expanding sphere needn't be tangent to it. And indeed, non-convex polyhedra obviously do exist which can't come to rest on any of their faces; any of the standard stellations of the dodecahedron or icosahedron is a good example. In that situation, replace the polyhedron by its convex hull (equivalent for stability purposes), and you find a stable orientation if not a literal face.

Another way to look at it is to observe that gravity will attempt to get the die's centre of mass as low as possible, so a face can only be unstable if there's some other orientation available with the CoM lower. So a polyhedron can only be astable if there's no face which places the CoM lowest – i.e. there must be an infinite sequence of faces each of which makes the CoM lower than the previous one. (You probably could construct such a shape, come to think of it; but (a) it wouldn't be a polyhedron in the strict sense any more, and (b) Bolzano-Weierstrass would dictate that there was some subsequence of those faces which converged to a limiting orientation, which would then necessarily be stable.)

[simont]

(Oops. Replace the polyhedron by its convex hull but with non-uniform density if necessary to put the CoM where it previously was, I meant. Of course the expanding-sphere proof works equally well if you've arranged for the CoM to be somewhere odd; just expand your sphere out from wherever it happens to be.)

[cartesiandaemon.livejournal.com]

That sounds right. Is there any chance we could call a stellated polyhedron a d0? It doesn't land on a face... :)