PS. More on the d7
Jul. 6th, 2007 02:48 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
jackhttp://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.
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.
no subject
Date: 2007-07-06 03:03 pm (UTC)Generallisable dn (n positive): n (or n+2 depending on definition) cylinder, cut of a bit from each end (so the dice looks like a hexagon from certain angles) sufficient to mean these sides are unstable => symmetrical dn.
Not sure if that description is understandable though....
no subject
Date: 2007-07-06 03:17 pm (UTC)Mostly clear, though I wasn't quite sure which solution you were proposing for the ends. I can think of several:
* Just cut the ends of flat and throw the pencil with enough force so it can't come to rest balanced on them.
* Have the faces bend at the end and all taper to a point.
* Stick two heptagonal pyramids on the ends, and it's not stable on them[1].
* Stick two heptagonal pyramids directly together.
[1] Actually, it doesn't matter if it *is* stable, it's like the previous example. There's actually 21 faces, but in three sets that clearly have equal chance within themselves, so it can still be used as a d7 if you label the faces repetitively.