Geometric and Combinatorial Properties of Nets in Plane and Higher-Dimensions




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This work belongs to a broad area of geometry of discrete integrable systems. The main objects of our study are nets which are configurations of affine subspaces limited to certain geometric and combinatorial constraints. For instance, confocal nets are characterized by their lines which are tangent to a given conic in plane. Similarly, incircular nets (IC nets) are congruences of straight lines in a plane having the property that every quadrilateral admits an inscribed circle. Checkerboard IC nets are a generalization of the IC nets in planar case and are defined by the property that every second elementary quadrilateral is circumscribed. One of the aims of this dissertation is to provide a new method of constructing confocal IC nets and confocal checkerboard IC nets which is based on integrable billiards. The second aim of this dissertation is to, following Böhm’s work, study of divisions of space into circumscribed cuboids. We give a proof of the following statement: a division of 3- 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. Then we generalize this statement to 4-dimensional case and prove it.



Conics, Spherical, Quadrics, Nets (Mathematics), Conic sections


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