Triangulations for the cube

Abstract

In this note we consider the problem of determining a minimal triangulation of I”, the n-dimensional cube. While the problem seems intrinsically interesting, our purpose in presenting it is motivated by the interest evinced in connection with the simplicial approximation of fixed points of continuous mappings [5, 71. Several algorithms for locating simplices which approximate fixed points have recently been given [l, 2, 3, 61. It is expected that by minimizing the number of simplices which fill a cube, the number of pivoting steps in the implementation of a fixed-point algorithm will generally be nearly minimal and that the resulting algorithm will generally perform with optimal efficiency. We consider here only triangulations with vertices of simplices coincident with vertices of the cube. We indicate techniques yielding triangulations of 13, 14, 15, consisting of 5, 16, 68 simplices of the respective dimensions. We show that 5 is the minimum number of simplices for a triangulation of I3 and that 16 is the minimum number for I4 subject to an additional hypothesis. We also give motivation for the conjecture that I” has a triangulation having (n ! + 29/2 simplices of dimension IZ.