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From: riks@mako.UUCP (Rik Smoody)
Newsgroups: net.math
Subject: Re: Polyhedral dice
Message-ID: <301@mako.UUCP>
Date: Mon, 17-Sep-84 18:06:57 EDT
Article-I.D.: mako.301
Posted: Mon Sep 17 18:06:57 1984
Date-Received: Tue, 25-Sep-84 04:18:57 EDT
References: <3639@decwrl.UUCP>
Organization: Tektronix, Wilsonville OR
Lines: 37



> 	I do not believe that it would be easy (or even possible) to
> construct a regular polyhedral die with unequal, though regular faces.
> However, fair dice can be made as long as all the FACES are identical
> to each other, although they may not be regular polygons.  The vertices
> and edges need not be all identical. (I do not know if there are any further
> requirements than this and a uniform material) An example of this is the 10
> .........
> 
> 						Mike Moroney
> 					..!decvax!decwrl!rhea!jon!moroney

The requirements are more stringent than that, and here's a counter
example of an unfair die, of uniform material, all faces equal.

Start with an octohedron.  Grab it by two opposing vertices
and pull it apart and give it a half twist.  Now connect each
of the vertices along the rift to the two closest ones in the other
half.   I find it hard to draw slanty lines in text.
This results in 8 new equilateral triangles connecting the old
square bases of the pyramidal caps.

Do this same process again.  Glue those triangles onto one of the caps,
and pull the other one away 'just far enough' to fill in with 8 more
new triangles. (The two caps will now be aligned again, and connected
by a lattice which has a square in the middle which is 45 degrees out
of line.

You can keep this process up for a while, if you are not already convinced
that the likelihood that this die land on one of the original triangles
is exceedingly small and best accomplished with glue!

Rik Smoody
Tektronix