Geometric properties of nets in plane and higher-dimensions

Modern discrete differential geometry sheds new light on many classical geometric construction and configuration. Recently, Akopyan and Bobenko have studied divisions of the Euclidean plane by lines into circumscribed quadrilaterals and called them incircular nets (IC nets) if every elementary quadrilateral possesses an inscribed circle. They pointed out to a classical paper of Wolfang Böhm, who has introduced the notion of a division of Euclidean plane by lines into circumscribed quadrilaterals. The aim of this talk is to present recent result about incircular nets and 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. We also generalize Böhm’s statement to n-dimensional case and prove it.