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From: pmartz@undies.dsd.es.com (Paul Martz)
Newsgroups: comp.graphics
Subject: Re: Questions and Answers about Platonic Solids
Keywords: platonic solids
Message-ID: <1991Mar8.224027.22703@dsd.es.com>
Date: 8 Mar 91 22:40:27 GMT
References: <2421@ria.ccs.uwo.ca> <2346@umriscc.isc.umr.edu>
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In article <2346@umriscc.isc.umr.edu>, mcastle@mcs213e.cs.umr.edu (Mike Castle {Nexus}) writes:
> In article <2421@ria.ccs.uwo.ca> pruss@ria.ccs.uwo.ca (Alexander Pruss) writes:
> >I recently received (indirectly) a set of co-ordinates for the seven platonic
>                                                                 ^^^^^
> Umm, aren't there only 5 platonic solids?? (According to Glassner, 6 when 
> counting the teapotahedron :-).
> 
> Most likely the 6th and 7th are symmetric in some way, but not true Platonic
> solids.
> 
> >solids originally posted (I think) by awpaeth@watcgl.waterloo.edu.
> 
> 
> -- 
> Mike Castle (Nexus) S087891@UMRVMA.UMR.EDU (preferred)       | XEDIT: Emacs
>                 mcastle@mcs213k.cs.umr.edu (unix mail-YEACH!)| on a REAL
> Life is like a clock:  You can work constantly, and be right | operating
> all the time, or not work at all, and be right twice a day.  | system. :->

I saved this original posting, but never looked at it close enough
(until now) to notice that, indeed, that last solid looks kind of
funky. Here it is for reference. The "..." don't intuitively map to
specific values. If anyone out there can explain it, please feel free.

> From: awpaeth@watcgl.waterloo.edu (Alan Wm Paeth)
> Newsgroups: comp.graphics
> Subject: Re: Polyhedra inscribed in unit sphere...
> Date: 25 Oct 90 15:31:42 GMT
> Organization: University of Waterloo
> 
> Here we go again... (this time 45 days elapsed between postings)
> 
> ------------------------------------------------------------------------------
> Coordinates for these and for their four-dimensional analogs were published by
> HSM Coxeter, first in 1948 in _Regular Polytopes_, pg. 52-53 (Methuen, London)
> and again in subsequent revisions; any/all are highly recommended reading. The
> table for (quasi) regular 3D polyhedra is transcribed below. I've posted this a
> few times already; perhaps a "frequently asked" entry is in order.
> 
> 
> 			PLATONIC SOLIDS
> 		(regular and quasi-regular variety,
> 		Kepler-Poinset star solids omitted)
> 
> The orientations minimize the number of distinct coordinates, thereby revealing
> both symmetry groups and embedding (eg, tetrahedron in cube in dodecahedron).
> Consequently, the latter is depicted resting on an edge (Z taken as up/down).
> 
>     SOLID            VERTEX COORDINATES
>     -----------      ----------------------------------
>     Tetrahedron      (  1,  1,  1), (  1, -1, -1), ( -1,  1, -1), ( -1, -1,  1)
>     Cube             (+-1,+-1,+-1)
>     Octahedron       (+-1,  0,  0), (  0,+-1,  0), (  0,  0,+-1)
>     Cubeoctahedron   (  0,+-1,+-1), (+-1,  0,+-1), (+-1,+-1,  0)
>     Icosahedron      (  0,+-p,+-1), (+-1,  0,+-p), (+-p,+-1,  0)
>     Dodecahedron     (  0,+-i,+-p), (+-p,  0,+-i), (+-i,+-p,  0), (+-1,+-1,+-1)
>     Icosidodecahedron(+-2,  0,  0), (  0,+-2,  0), (  0,  0,+-2), ...
>                      (+-p,+-i,+-1), (+-1,+-p,+-i), (+-i,+-1,+-p)
> 
>     with golden mean: p = (sqrt(5)+1)/2; i = (sqrt(5)-1)/2 = 1/p = p-1
> ------------------------------------------------------------------------------
> 
> The poster wanted a circumscribing (unit) sphere. Just pick a vertex and
> calculate its length (to the origin) and you have R, that sphere's radius.
> Normalize (divide all coordinates by R) and the solids are contained by a
> unit sphere.
> 
>     /Alan Paeth
>     Computer Graphics Laboratory
>     University of Waterloo
-- 

   -paul	pmartz@dsd.es.com
		Evans & Sutherland