Division of n -Dimensional Euclidean Space into Circumscribed n-Cuboids

Abstract

In 1970, Böhm formulated a three-dimensional version of his two-dimensional theorem that a division of a plane by lines into circumscribed quadrilaterals necessarily consists of tangent lines to a given conic. Böhm did not provide a proof of his three-dimensional statement. The aim of this paper is to give a proof of Böhm’s statement in three dimensions that a division of three-dimensional Euclidean space by planes into circumscribed cuboids consists of three families of planes such that all planes in the same family intersect along a line, and the three lines are coplanar. Our proof is based on the properties of centers of similitude. We also generalize Böhm’s statement to the four-dimensional and then n-dimensional case and prove these generalizations.