Tight fans and their canonical scalings

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.