The Local Theorem for Monotypic Tilings

Abstract

A locally finite face-to-face tiling $\cal T$ of euclidean $d$-space ${\Bbb E}^d$ is monotypic if each tile of $\cal T$ is a convex polytope combinatorially equivalent to a given polytope, the combinatorial prototile of $\cal T$. The paper describes a local characterization of combinatorial tile-transitivity of monotypic tilings in ${\Bbb E}^d$; the result is the Local Theorem for Monotypic Tilings. The characterization is expressed in terms of combinatorial symmetry properties of large enough neighborhood complexes of tiles. The theorem sits between the Local Theorem for Tilings, which describes a local characterization of isohedrality (tile-transitivity) of monohedral tilings (with a single isometric prototile) in ${\Bbb E}^d$, and the Extension Theorem, which gives a criterion for a finite monohedral complex of polytopes to be extendable to a global isohedral tiling of space.