Tight fans and their canonical scalings

TL;DR

This paper introduces tight fans, explores their relation to parallelotope tilings, and proves key theorems linking fan properties to canonical scalings and polytopality in Euclidean space.

Contribution

It defines tight fans based on local symmetry, proves their connection to parallelotope tilings, and establishes conditions for the existence of canonical scalings.

Findings

Tight fans are characterized by local symmetry in face tilings.

A fan of cells meeting at a face in a parallelotope tiling is a tight fan.

A fan admits a canonical scaling if and only if it is polytopal.

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.