Path: utzoo!attcan!uunet!zaphod.mps.ohio-state.edu!usc!ucsd!pacbell.com!att!att!watmath!watserv1!watcgl!awpaeth From: awpaeth@watcgl.waterloo.edu (Alan Wm Paeth) Newsgroups: comp.graphics Subject: Re: Polyhedra inscribed in unit sphere... Message-ID: <1990Oct25.153142.29468@watcgl.waterloo.edu> Date: 25 Oct 90 15:31:42 GMT References: Organization: University of Waterloo Lines: 40 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