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From: moroney@jon.DEC
Newsgroups: net.math
Subject: Re: Polyhedral dice
Message-ID: <3702@decwrl.UUCP>
Date: Fri, 21-Sep-84 14:41:46 EDT
Article-I.D.: decwrl.3702
Posted: Fri Sep 21 14:41:46 1984
Date-Received: Tue, 25-Sep-84 21:52:03 EDT
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Organization: DEC Engineering Network
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>> 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
>> .........
 

>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.
.......the likelihood that this die land on one of the original triangles
is exceedingly small...

You are right.. There is another important requirement - the angle between
any 2 faces must be the same, or each face must have an identical environment
around it to all the other faces.

So far the only polyhedra which meet these that I know of are the 5 regular
polyhedra, the "30" in role-playing games (Anyone know the official name
for this?) and the 12-sided rhomboid I mentioned earlier.

 						Mike Moroney
 					..!decvax!decwrl!rhea!jon!moroney