Dissection of the hypercube into simplexes

Abstract

A generalization of Sperner’s Lemma is proved and, using extensions of p-adic valuations to the real numbers, it is shown that the unit hypercube in n dimensions can be divided into m simplexes all of equal hypervolume if and only if m is a multiple of n!. This extends the corresponding result for n = 2 n = 2 of Paul Monsky.