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 socalled 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 ≥ 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!(e − 2).

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