From:zare@cco.caltech.edu (Douglas J. Zare)Newsgroups:sci.mathSubject:Re: On simplex/hyperplane intersectionDate:3 Jan 1996 00:33:08 GMTOrganization:California Institute of Technology, Pasadena

James C. Evans <jce@seaice.geol.scarolina.edu> wrote: >Suppose that a "solid" simplex of dimension n that has of n+1 >vertices, each of which is a point having n coordinates, is >intersected by a hyperplane of integer dimension m, 0 < m < n, >namely > sum_{i=0}^m a_i * x_i = b . >By solid, I mean that the simplex should be regarded as including >its interior. > >Is it true that the intersection is either null, or is another >solid simplex of dimension p, where 0<p < m and where a point >and a line segment here are defined as degenerate simplexes of >dimension 0 and 1, respectively? >[...] No, you can get a square by cutting a tetrahedron in half such that two vertices are "above" the plane and two are below. In slightly more generality, if there are n points above the plane and m below, then the intersection is the Cartesian product of an n-point simplex and an m-point simplex. For example, (n,m)=(2,3) yields a triangular prism. Note that there are nm vertices of this figure, yet in the most symmetric case there are only 2 distances between the vertices. If there are p points in the plane, n above, and m below, then the result is the p-times iterated cone over the product of n-point and m-point simplices. For example, when (p,n,m) = (1,2,2), the result is a square-based pyramid; when (p,n,m) = (2,2,2), the result is a hyper pyramid whose base is a square-based pyramid. This even makes sense when one or more of p, n, and m equal 0. The proofs are straightforward, so I hope I did them correctly: First, note that the combinatorial type is determined by p, n, and m. Second, use the coordinatization of an k-point simplex as nonnegative real ordered k-tuples which sum to 1. For the p=0 case, it suffices to consider the plane defined by setting the sum of the first n coordinates to 1/2; this is also implies that the sum of the last m coordinates is 1/2, hence the intersection is the Cartesian product of 1/2-size n-point and m-point simplices. The case of p>0 is similar. This is a really neat problem. I'll have to think about the higher co-dimension cases and cubes, cross-polytopes, and hemi-cubes instead of simplices. I'll also post to the Geometry Forum's geometry.puzzles, geometry.college, and/or goemetry.research (http://forum.swarthmore.edu/~sarah/HTMLthreads/index.html), but I don't see how to link those to posts to sci.math. Douglas Zare http://www.cco.caltech.edu/~zare