A Local Characterization of Combinatorial Multihedrality in Tilings

TL;DR

This paper provides a local criterion using centered coronas to identify combinatorially multihedral tilings in Euclidean space, extending previous results on tile-transitivity.

Contribution

It introduces a local characterization of combinatorially multihedral tilings, generalizing the earlier Local Theorem for Monotypic Tilings.

Findings

Provides a local criterion for multihedrality

Generalizes the Local Theorem for Monotypic Tilings

Applicable to face-to-face tilings by convex polytopes

Abstract

A locally finite face-to-face tiling of euclidean d-space by convex polytopes is called combinatorially multihedral if its combinatorial automorphism group has only finitely many orbits on the tiles. The paper describes a local characterization of combinatorially multihedral tilings in terms of centered coronas. This generalizes the Local Theorem for Monotypic Tilings, established in an earlier paper, which characterizes the case of combinatorial tile-transitivity.