TOPAZ:shallit@ucbvax.UUCP (07/22/83)
I received many replies to my question about how to compute the XYZ co-ordinates for the regular dodecahedron and icosahedron. Thanks to all who replied. Some of the replies are summarized here: ggs@ulysses.UUCP suggested an approximation approach that figured out the correct points by successive rotations. He also sent a FORTRAN program to compute the draw co_ordinates. Lee Dickey suggested the book "Regular Polytopes" by H. S. M. Coxeter. Bruce Cohen looked up the co-ordinates in this book and provided the ones for the icosahedron. In the meantime, I did a little work and came up with another reference and a tricky way to get the co-ordinates for the dodecahedron. In Audrey Tam's Ph. D. thesis "Optimal Choice of Directions for the Reconstruction of an Object from a Finite Number of its Plane Integrals", she gives the following co-ordinates for an icosahedron centered at (0,0,0), inscribed in a sphere of radius 1: Let p^2 = (5+sqrt(5))/10; q^2 = (5-sqrt(5))/10. Then the 12 vertices of the icosahedron are: ( +-p, +-q, 0) ; ( 0, +-p, +-q ) ; ( +-q, 0, +-p ). She got this from an article by McLaren in Math. Comp. in 1963. Now that we have the vertices, how do we determine the edges? Well, there are 2!12 ways of choosing pairs of points; it is easy to determine the distance between each point. Now choose the edges as those connecting points of minimum distance. Now that we have the edges, how do we determine faces? Well, we can easily compute all possible ways to choose the 3 points that define a face. Now check to see that the 3 implied edges in such a choice really exist in the edge set. Now remove duplicates up to a) cycling and b) reversal. You're done. Now we can compute the vertices for the dodecahedron, because the vertices of a dodecahedron are just the centroids (center of mass) of the faces of the icosahedron! If you want the points to lie on the sphere of radius 1, you will have to normalize the XYZ co-ordinates. The same sort of trick works with the cube (whose vertices are easy to describe: just (+-a, +-a, +-a) where a = sqrt(1/3)) and the octahedron. However, the tetrahedron is self-dual. I have APL programs to compute the co-ordinates, edges, and faces of all 5 regular solids and will be glad to (U. S.) mail them to anyone who's interested. /Jeff Shallit, UC Berkeley.