There are only two nonobtuse binary triangulations of the unit n-cube

Abstract

Triangulations of the cube into a minimal number of simplices without additional vertices have been studied by several authors over the past decades. For 3⩽n⩽7 this so-called simplexity of the unit cube In is now known to be 5,16,67,308,1493, respectively. In this paper, we study triangulations of In with simplices that only have nonobtuse dihedral angles. A trivial example is the standard triangulation into n! simplices. In this paper we show that, surprisingly, for each n⩾3 there is essentially only one other nonobtuse triangulation of In, and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than n!(e−2).