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Triangulations of the cube into a minimal number of simplices withoutadditional 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 nowknown to be $5,16,67,308,1493$, respectively. In this paper, we studytriangulations of $I^n$ with simplices that only have nonobtuse dihedralangles. A trivial example is the standard triangulation into $n!$ simplices. Inthis paper we show that, surprisingly, for each $n\geq 3$ there is essentiallyonly one other nonobtuse triangulation of $I^n$, and give its explicitconstruction. The number of nonobtuse simplices in this triangulation is equalto the smallest integer larger than $n!({\rm e}-2)$.

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