There Is No Face-to-Face Partition of R5 into Acute Simplices

We prove that a point in the Euclidean space R⁵ cannot be surrounded by a finite number of acute simplices. This fact implies that there does not exist a face-to-face partition of R⁵ into acute simplices. The existence of an acute simplicial partition of R^d for d > 5 is excluded by induction, but for d = 4 this is an open problem. Dedicated to Prof. Miroslav Fiedler on the occasion of his 80th birthday