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||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.
|Publisher:||Springer Link||Project:||Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems|
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checked on Jan 31, 2023
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