Dissection of the hypercube into simplexes

Abstract

A generalization of Sperner’s Lemma is proved and, using extensions of <italic>p</italic> -adic valuations to the real numbers, it is shown that the unit hypercube in <italic>n</italic> dimensions can be divided into <italic>m</italic> simplexes all of equal hypervolume if and only if <italic>m</italic> is a multiple of <italic>n</italic> !. This extends the corresponding result for <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 2"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n = 2</mml:annotation> </mml:semantics> </mml:math> </inline-formula> of Paul Monsky.