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PROPERTIES OF BÖRÖCZKY?S CONSTRUCTION
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Of a special interest are tilings in hyperbolic n-space . It is natural to extend the study of tiling problems to the hyperbolic plane as well as hyperbolic spaces of higher dimension. In this work we consider Karoly Böröczky tilings in hyperbolic space in arbitrary dimension, study some properties and some useful consequences of this Böröczky's construction. In the given work it will be shown, that Böröczky tiling has one more remarkable property using them it is simple to make examples of not face-to-face tilings of the hyperbolic n-dimensional space composed of congruent (equal), convex and compact polyhedral tiles. Additionally, these tilings also cannot be transformed in isohedral tilings using polytopes permutation as well. The obtained tilings of n- dimensional hyperbolic space are important as well, due to the fact that the examples of isohedral tilings of hyperbolic n-dimensional space by compact polyhedral tiles are not yet constructed. The proposed construction could be considered as well as constructive demonstration related to the theorem of existence of not face-to-face tilings of hyperbolic n - dimensional space by equal, convex and compact polytopes.
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