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arXiv · 1603.01873

Tight fans and their canonical scalings

Andrei Gavrilyuk
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Abstract

A new class of full fans in an euclidean space - tight fans - is introduced. Such fans are defined using a property of local symmetry in a face of a tiling. Tight fans are related to the theory of parallelotopes in an euclidean space. A theorem is proved that a fan of cells meeting in a given face of a tiling by parallelotopes is a tight fan. A new proof was given for a theorem by Delone on 5 combinatorial types of parallelotopes meeting in a common face of codimension 3. Canonical scalings of an euclidean space tiling are special functions defined on hyperfaces of the tiling. Existance of such functions for a given tiling is known to be related to existance of a generatix of a tiling. Generatix is a polyhedral surface with an orthogonal projection coinciding with a given tiling. A theroem is proved that a fan has a canonical scaling iff it is polytopical.

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Andrei Gavrilyuk. 2016-03-06. Tight fans and their canonical scalings. https://arxiv.org/abs/1603.01873

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