fontenot@rice.edu (Dwayne Jacques Fontenot) (09/09/90)
Hello. I have been searching the standard references (Foley & van Dam, Rogers, etc.) and have not been able to find and discussion of drawing 3d regular solids ; pyramid, cube, octahedron, icosahedron, dodecahedron, etc... It seems that the only possible regular polygonal solids that can exist have numbers of sides equivalent to members of the Pythagorean series. I want to know why this is (as well as how to draw them ; I have access to a 3d graphics package). Ideally I could get pointers to proofs and algorithms. Thank you for your time. Dwayne Fontenot
dbc@cs.brown.edu (Brook Conner) (09/09/90)
Dwayne, The Platonic solid are the only regular solids because they are :) Seriously, I'm sure this is a result of some result in topology somewhere (although having only a passing acquaintance with topology stemming from working under a topologist (John Hughes of Foley, van Dam, Feiner, and Hughes) I have yet to see this proof for myself, so I can't offer pointers to it) But the real point of this post is how to draw them, which is discussed in detail in "Jim Blinn's Corner" in IEEE Computer Graphics and Applications, November 1987, volume 7 number 11, page 62. Blinn presents sets of coords that form a tetra hedron, octahedron, dodecahedron, and icosahedron (the cube is left as an exercise for the reader :) using only 1.0, 0.0 and 1.618034 (i.e. (1 + sqrt(5))/2 ) and their additive inverses. Brook Brook Conner | Klacktoveedsedstene Brown Computer Graphics | Fortune sez: Brook's Law -- Adding manpower to a late dbc@cs.brown.edu | software project makes it later uunet!brunix!dbc dbc@browncs.bitnet Box 4013 Brown U Prov RI 02912
awpaeth@watcgl.waterloo.edu (Alan Wm Paeth) (09/11/90)
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