pruss@ria.ccs.uwo.ca (Alexander Pruss) (03/05/91)
I recently received (indirectly) a set of co-ordinates for the seven platonic solids originally posted (I think) by awpaeth@watcgl.waterloo.edu. I wrote some clever code that changed these co-ordinates into a more useful form for people like me who actually want to look at the solids - the code takes the different vertex co-ordinates, and figures out which are connected to form edges and which edges connect to form faces. Then your favourite 3d graphic package can be easily used to look at them. The reason I am posting is first to let people know that this info is available (if they care ;-) and second is to ask about the 7th solid in the list, the Icosidodecahedron. The co-ordinate list of this solid contains an ominous looking ... which doesn't seem to obviously translate into extra vertices. Currently it has 30 vertices (I think) but my program decides that this means there should be some fractional (2.2 or something) number of edges in a face which isn't right. Are there some more vertices? How many edges/faces should it have and are all the edges the same length (my program assumes all edges are the same length to work out # edges, # faces, # edges/face etc. which is true for all other platonic solids). What gives? pat .
mcastle@mcs213e.cs.umr.edu (Mike Castle {Nexus}) (03/08/91)
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. :->
pmartz@undies.dsd.es.com (Paul Martz) (03/09/91)
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