Authors: Dragović, Vladimir 
Ranomenjanahary, Roger Fidèle
Affiliations: Mathematical Institute of the Serbian Academy of Sciences and Arts 
Title: Division of n -Dimensional Euclidean Space into Circumscribed n-Cuboids
Journal: Proceedings of the Steklov Institute of Mathematics
Volume: 310
Issue: 1
First page: 137
Last page: 147
Issue Date: 4-Dec-2020
Rank: M22
ISSN: 0081-5438
DOI: 10.1134/S0081543820050119
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.
Publisher: Springer Link
Project: Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems 

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