bk0y+@andrew.cmu.edu (Brian Christopher Kircher) (10/24/90)
Does anyone out there have data-sets (i.e. vertex, edge lists) for a dodecahedron and a icosahedron inscribed in the unit sphere? It looks like some wickedly evil math to compute the vertices and I would rather not work it out if it could be avoided. Much thanks in advance... Brian Kircher
philip@beeblebrox.dle.dg.com (Philip Gladstone) (10/24/90)
In article <gb9_r1m00WB4MKFkhQ@andrew.cmu.edu> bk0y+@andrew.cmu.edu (Brian Christopher Kircher) writes:
Does anyone out there have data-sets (i.e. vertex, edge lists) for a
dodecahedron and a icosahedron inscribed in the unit sphere? It looks
like some wickedly evil math to compute the vertices and I would
rather not work it out if it could be avoided.
The vertices for dodecahedron and icosahedron are not that difficult
to compute. I have a friend whose MSc project was to draw all 85 of
the regular polyhedra. That is tricky I can assure you.
--
Philip Gladstone philip@dle.dg.com
Development Lab Europe C=gb/AD=gold 400/PR=dgc/O=dle
Data General, Cambridge /SN=gladstone/GN=philip
England. +44 223-67600jroth@allvax.enet.dec.com (Jim Roth) (10/25/90)
In article <PHILIP.90Oct24114118@beeblebrox.dle.dg.com>, philip@beeblebrox.dle.dg.com (Philip Gladstone) writes... >In article <gb9_r1m00WB4MKFkhQ@andrew.cmu.edu> bk0y+@andrew.cmu.edu (Brian Christopher Kircher) writes: > > Does anyone out there have data-sets (i.e. vertex, edge lists) for a > dodecahedron and a icosahedron inscribed in the unit sphere? It looks Try looking up a book on geodesic domes (there are several such books around...) Another book worth looking up is called "Mathematical Models" by Cundy and Rollett (Oxford) - it has many of the solids. Another author to look for is H. S. M. Coxeter, a geometer who has quite a number of fascinating books at several mathematical levels. Then you will know such trivia as what a great rhombicosidodecahedron is :-) - Jim
truett@cup.portal.com (Truett Lee Smith) (10/25/90)
There was an article by Blinn in the IEEE Graphics journal during about a year ago which showed how to derive the Platonic solids by hand using a few simple tricks. It's really neat. The dodecagon, as I remember, was formed by the vertices of three golden rectangles intersecting at the origin. Truett Smith Sunnyvale, CA E-mail: truett@cup.portal.com OR truett@tdd.sj.nec.com
v134kkut@ubvmsd.cc.buffalo.edu (David W Tinklepaugh) (10/25/90)
In article <gb9_r1m00WB4MKFkhQ@andrew.cmu.edu>, bk0y+@andrew.cmu.edu (Brian Christopher Kircher) writes... >Does anyone out there have data-sets (i.e. vertex, edge lists) for a >dodecahedron and a icosahedron inscribed in the unit sphere? It looks >like some wickedly evil math to compute the vertices and I would >rather not work it out if it could be avoided. I would just like to point out that this question is IDENTICAL to: How do you tesselate a sphere How to find measurements to build a geodesic dome How to represent a ball as a group of similar polygons other than the globe-longitude-latitude method. oh, one answer is in the "read this before you post" post listed under #15, tesselation. (I tried to investigate this myself once and found my info in the architecture library of all places. I didn't even know who Buckmeister Fuller was :) ) ______________________________________________________________________ / / \ / / -Dave- U. of Buffalo / Wrap it up, I'll take it! \ / / v134kkut@ubvmsa (bitnet) / -Fabulous Thunderbirds \ / /_______________________________/____________________________________\/
awpaeth@watcgl.waterloo.edu (Alan Wm Paeth) (10/25/90)
Here we go again... (this time 45 days elapsed between postings)
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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
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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