janc (01/20/83)
I'm interested in computing the probabilities with which an
irregularly shaped polyhedron lands on its various faces when
rolled.
This problem seems to have a certain "classical flavor" to it,
and I'm sure work has been done on it, but I haven't been able to
find anything on the problem.
Can anyone point me toward any literature on estimating the
odds for irregular dice, or on the physics of rolling objects?
Thank you.
-- Jan WolterJGA@MIT-MC (01/21/83)
From: John G. Aspinall <JGA @ MIT-MC>
Date: 20 Jan 83 9:36:24-PST (Thu)
From: ihnss!ihldt!ll1!sb1!mb2b!uofm-cv!janc@Berkeley.arpa
Article-I.D.: uofm-cv.130
Received: from Usenet.uucp by SRI-UNIX.uucp with rs232; 21 Jan 83 2:06-PST
I'm interested in computing the probabilities with which an
irregularly shaped polyhedron lands on its various faces when
rolled.
I don't know of any literature references to this, but it sounds like
a fun problem. To start with you could assign to each face a
probabililty proportional to the solid angle subtended by the face, as
seen from the center of mass (com).
But our die rolls when it lands, and some faces may not even be stable
to sit upon. So consider the potential energy of all possible
orientations of the die. This is the distance from the com to the
surface at that orientation. We can regard the rolling and stopping
of the die as the dynamics of a point on that surface. The potential energy
is given by the height of the point, while the kinetic energy has
to be found from some sort of energy conservation conditions.
But here's the hard part - we have to include dissipation of the
energy in the model, or else the die will never roll to a stop. And
it's easy to see that the form of the dissipation will affect the
probabilities quite drastically -- throw the die on a sticky surface
(high dissipation) and you greatly increase the probabilities of
landing on a small face.
Once you've got the dissipation model, then all you have to do (heh!)
is a integral over all the possible initial conditions. This could be
a Monte-Carlo technique or something more analytic.
John Aspinall.
P.S. Despite the glibness of this reply, I really do find the subject
interesting. I'd like to hear more about it.