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From: rjnoe@ihlts.UUCP (Roger Noe)
Newsgroups: net.math
Subject: Polyhedral dice
Message-ID: <554@ihlts.UUCP>
Date: Wed, 12-Sep-84 09:45:44 EDT
Article-I.D.: ihlts.554
Posted: Wed Sep 12 09:45:44 1984
Date-Received: Fri, 14-Sep-84 06:42:53 EDT
Organization: AT&T Bell Labs, Naperville, IL
Lines: 32

Everyone's familiar with the ordinary six-sided die, the regular cube.
Many are familiar with dice in the shape of the other four regular polyhedra,
the tetrahedron (4 triangular faces), octahedron (8 triangular faces),
dodecahedron (12 pentagonal faces) and the icosahedron (20 triangular faces).
If we assume all these are fair dice (i.e. absolutely constant density of
the solid material which makes up the dies and perfectly sharp edges), the
notion of symmetry dictates that each face of one of these dice is equally
likely to turn up (or down) as is any of the other faces of the same die.
But what about fair polyhedra which are not quite regular?  I'm not up on
all of the terminology in this area, so I would appreciate others clarifying
some of this for me.  Here's what I'm aiming at:
	1. Each of the faces of these "semi-regular" polyhedra is a regular
	   polygon but not all the faces are the same.  For instance, we
	   could have a polyhedron with squares and equilateral triangles
	   making up the faces.
	2. The polyhdedra are all convex, so that the relationship
		faces + vertices = edges + 2
	   applies.
	3. Each vertex is equivalent to each of the other vertices in terms
	   of the number of edges incident and the pattern of faces around
	   it (allowing for rotation).
Symmetry again would indicate that all the faces of one type on the polyhedron
would have equal probabilities of ending on bottom if the die is rolled.  I am
unsure of how to go about finding this probability.  It would seem to have
something to do with the area of the face and the dihedral angles about that
face.  I would expect it to be independent of the speed at which the die is
rolled.  Anyone have any information on this topic or any ideas how to go about
analytically determining the probabilities of rolling one face or another?
-- 
"It's only by NOT taking the human race seriously that I retain what
    fragments of my once considerable mental powers I still possess."
Roger Noe			ihnp4!ihlts!rjnoe