polytopes and cross products

*To*: mathgroup at smc.vnet.net*Subject*: [mg19290] polytopes and cross products*From*: Russell Towle <rustybel at foothill.net>*Date*: Thu, 12 Aug 1999 22:34:46 -0400*Sender*: owner-wri-mathgroup at wolfram.com

Hi all, In prologue, I am using Mathematica to construct various regular and semi-regular polytopes in n-dimensional space, especially in four dimensions. I have spent many hundreds of hours on a notebook which constructs these 4-polytopes, makes solid shadows of them, and projections into various sub-spaces, and even hidden-detail-removed projections of 4-polytopes into 3-spaces. In order to bring this notebook to a point where it can be placed on MathSource, I must solve one last nagging problem. First, a 4-polytope is bounded by polyhedra. Take the regular 120-cell, {5,3,3}. It is bounded by 120 Platonic pentagonal dodecahedra. In the 4-space, supposing the dodecahedra are opaque, an observer would only see half, or somewhat less than half, of them. In order to find which ones are visible from an arbitrary viewpoint in the 4-space, I must find the cross product for each dodecahedron: the vector perpendicular to its own particular 3-space. My cross product function "Cross4" requires four 4-vectors as input. So I will write, for instance, for a dodecahedron named "dodeca", Cross4[ dodeca[[ {1,5,11,17} ]] ]. I loop through all 120 dodecahedra in such a way. I find by trial-and-error that, say, the 1st, 5th, 11th, and 17th vertices of any of the 120 dodecahedra span their respective 3-spaces. And I find by trial-and-error that, say, the 1st, 2nd, 3rd, and 4th vertices of the 120 dodecahedra *do not always* span the 120 3-spaces. As it stands now, once I have picked a good set of four indices, I can find the visible dodecahedra in a second or two. What I need, though, is a fast and reliable way to pick out subsets of four vertices from *any* polyhedron in a 4-space, such that the four vertices *span the 3-space of the polyhedron*. For it may well happen, as in the case of the dodecahedra, that some four of the 20 vertices of a polyhedron are all in a single plane (a single polygon) and thus do not span the 3-space of the polyhedron. In summary, how can I use Mathematica to select a set of four 4-vectors which span a 3-space in the 4-space? Russell Towle Box 141 Dutch Flat, CA 95714 (530) 389-2872