Abstract

A unit N-simplex is a regular N-dimensional simplex whose N + 1 vertices lie on the unit sphere in R^N. The equidistant vertices are spread evenly over the sphere, and thus are useful in applications that require a representative sample of points on a sphere (such as N-dimensional integration). It is always possible to choose coordinate axes such that the set of vertices is invariant under all permutations of the coordinates. This paper gives two different mathematical constructions of coordinate permutation invariant unit N-simplexes in N-dimensional space. The first construction method involves translation followed by rotation in N + 1-dimensional space, while the second requires only a single translation in N-dimensional space. The development of these constructions will begin with a unit 2-simplex (i.e., equilateral triangle) in R², with visualizations in GeoGebra. These arguments are then generalized to the N-dimensional case for any integer N ≥ 2. The constructions also provide a simple derivation for the formula for the length of the edges of a unit N-simplex, from which may be derived the angle between any two vertex vectors. These constructions and their applications involve an interesting mixture of geometry, algebra, and analysis. The GeoGebra visualizations for 2- and 3-dimensional simplexes are available online. For readers who want to cut to the chase, formulas for unit N-simplex vertices and edge lengths are summarized in the last section.

Keywords: Simplex, Permutation, Coordinates, Invariance, Numerical integration, GeoGebra