From: utzoo!decvax!harpo!seismo!hao!hplabs!sri-unix!JGA@MIT-MC
Newsgroups: net.physics
Title: irregular dice--info wanted
Article-I.D.: sri-arpa.308
Posted: Fri Jan 21 09:40:00 1983
Received: Mon Jan 24 01:15:21 1983

From:  John G. Aspinall 

    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.