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Division of \(n\)-Dimensional Euclidean Space into Circumscribed \(n\)-Cuboids

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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.

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Acknowledgments

The authors would like to thank Professor Frank Konietschke for his help in translating Böhm’s work [4].

Funding

The research was partially supported by the Serbian Ministry of Education, Science and Technological Development (project no. 174020 “Geometry and Topology of Manifolds, Classical Mechanics and Integrable Dynamical Systems”) and by The University of Texas at Dallas.

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Authors and Affiliations

  1. Department of Mathematical Sciences, The University of Texas at Dallas, 800 W. Campbell Rd., FO 35, Richardson, TX, 75080, USA

    Vladimir Dragović & Roger Fidèle Ranomenjanahary

  2. Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000, Belgrade, Serbia

    Vladimir Dragović

Authors
  1. Vladimir Dragović
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  2. Roger Fidèle Ranomenjanahary
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Correspondence to Vladimir Dragović.

Additional information

To Valery Vasil’evich Kozlov on the occasion of his anniversary

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Dragović, V., Ranomenjanahary, R.F. Division of \(n\)-Dimensional Euclidean Space into Circumscribed \(n\)-Cuboids. Proc. Steklov Inst. Math. 310, 137–147 (2020). https://doi.org/10.1134/S0081543820050119

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  • DOI: https://doi.org/10.1134/S0081543820050119

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