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

TL;DR

This paper proves that for each dimension n ≥ 3, there are only two nonobtuse binary triangulations of the unit n-cube, providing an explicit construction for the second triangulation and analyzing its properties.

Contribution

It establishes the uniqueness of a second nonobtuse triangulation of the unit n-cube and provides its explicit construction, expanding understanding of cube triangulations.

Findings

There are exactly two nonobtuse binary triangulations of the unit n-cube for n ≥ 3.

The second triangulation has a number of simplices equal to the smallest integer greater than n!({ m e}-2).

The standard triangulation into n! simplices is one of the two nonobtuse triangulations.

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 $3\leq n\leq 7$ this so-called simplexity of the unit cube $I^n$ is now known to be $5,16,67,308,1493$, respectively. In this paper, we study triangulations of $I^n$ 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 $n\geq 3$ there is essentially only one other nonobtuse triangulation of $I^n$, and give its explicit construction. The number of nonobtuse simplices in this triangulation is equal to the smallest integer larger than $n!({\rm e}-2)$.