(no subject)

It's always good to keep a record of error messages and the bugs that caused them, if they're the sort of misconfiguration bugs that are likely to crop up all the time. Then next time, you don't have to try to remember or try to rediagnose, you can just go to your previous self and ask "Hey, what caused this last time? Oh, of course."

The *last* time the zip on my coat jammed, it took about two hours sitting on a sofa to fix. *This* time I got it loose in two minutes by the side of the road, without taking it off. It was exactly the same, except that 98% of the job was thinking, which I'd already done.

So, contrary to all expectation, I'm *not* living in a maths joke, since I actually went ahead and did that extra 2% effort to actually free it, rather than simply calling the job done because I knew I *could* do it.

(The first time I did have to do it because I couldn't be sure until I'd tried it.)

There really IS a polyhedral d1

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! :)

PS. More on the d7

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.

More on the d7

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.