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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 Rd for d 5 is excluded by induction, but for d = 4 this is an open problem.

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