Newsgroups: comp.graphics
Path: utzoo!utgpu!watserv1!watcgl!awpaeth
From: awpaeth@watcgl.waterloo.edu (Alan Wm Paeth)
Subject: Re: Need references on drawing 3d regular polygonal solids
Message-ID: <1990Sep10.181045.4074@watcgl.waterloo.edu>
Organization: University of Waterloo
References: <1990Sep9.095906.26612@rice.edu>
Date: Mon, 10 Sep 90 18:10:45 GMT
Lines: 32

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, 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
------------------------------------------------------------------------------

   /Alan Paeth
   Computer Graphics Laboratory
   University of Waterloo