Simplexes, An Introduction

Introduction

At an intuitive level, a simplex is an n dimensional pyramid, which may be tilted or stretched. Like a closed ball, a simplex includes its interior. I'll provide a technical definition on the next page.

The "standard" simplex is defined by the unit vectors. Thus a 3 dimensional simplex has 4 corners, the origin and the ends of the 3 unit vectors. This looks like a pyramid with one corner squared off. A 4 dimensional simplex has 5 corners, and so on.

Another convenient formula for an n-simplex is a set of n+1 points in n dimensions, such that each is a fixed distance d from all the others. This is a line segment in 1 dimension, an equilateral triangle in 2 dimensions, a tetrahedron in 3 dimensions, and so on. Use induction to show such a simplex exists for each n. This formulation has a symmetry that all others lack - hence it is called the "symmetric simplex". Like a perfect multidimensional die, the shape may be rotated onto itself in any of (n+1)!/2 ways, giving a rotation group of A_n+1. If you include reflections, all (n+1)! permutations are possible, as indicated by the symmetric group S_n+1 on the vertices. Thus the tetrahedron exhibits 12 rotations, or 24 if you allow reflections through a mirror.

The symmetric simplex has another representation, if you don't mind working in n+1 dimensions. Consider the points in n+1 space whose coordinates are nonnegative, and sum to 1. For example, an equilateral triangle is bounded by the xy, xz, and yz planes in 3 space. The vertices of this shape are the ends of the unit vectors in n+1 space. Since every pair of vertices is sqrt(2) units apart, we haven't lost our symmetry. As we shall see, a natural map takes this shape, i.e. the front face of the standard simplex in n+1 dimensions, onto the symmetric simplex in n dimensions.

Before we proceed, I have to talk about plurals. One vertex leads to two vertices, and one matrix leads to two matrices. So - are there two simplexes or two simplices? According to Merriam Webster, either is correct. However, we will also define a construct called a complex, consisting of several simplexes pasted together. Are there then two complexes, or two complices? The latter is awkward, and I have never seen it in a text book or heard it in a lecture. Therefore, I'm going to stick with complexes, and to be consistent, I will let simplexes stand as the plural of simplex.