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.

中文翻译：

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将$$ n $$-欧几里德空间划分成外接的$$ n $$-长方体

摘要

1970年，伯恩（Böhm）提出了他的二维定理的三维形式，即平面将线段划分为外接四边形必须将给定圆锥形的切线组成。伯姆没有提供他的三维陈述的证明。本文的目的是从三个维度上证明伯姆的说法：将三维欧几里德空间按平面划分为外接长方体由三个平面族组成，使得同一族中的所有平面沿一条直线相交，并且这三条线是共面的。我们的证明是基于相似中心的性质。我们还将博姆的陈述推广到四维然后\（n \）维的情况，并证明这些推广。