Tessellations and Positional Representation

TL;DR

This paper establishes a correspondence between inflation-based substitution tilings and positional number systems, introducing a generalized framework for classifying tiles and analyzing tiling properties across dimensions.

Contribution

It generalizes inflationary tessellations to equivalence classes of tiles and links tiling inflation multipliers to algebraic numbers, enabling cross-dimensional analysis.

Findings

Multiplier for inflationary tilings is algebraic.

Equivalence under measure simplifies tiling property analysis.

Illustrates correspondence with Penrose, Ammann, and Taylor tilings.

Abstract

The main goal of this paper is to define a 1-1 correspondence between between substitution tilings constructed by inflation and the arithmetic of positional representation in the underlying real vector space. It introduces a generalization of inflationary tessellations to equivalence classes of tiles. Two tiles belong to the same class if they share a defined geometric property, such as equivalence under a group of isometries, having the same measure, or having the same `decoration'. Some properties of ordinary tessellations for which the equivalence relation is congruence with respect to the full group of isometries are already determined by the weaker relation of equivalence with respect to equal measure. In particular, the multiplier for an inflationary tiling (such as a Penrose aperiodic tiling) is an algebraic number. Equivalence of tiles under measure facilitates the investigation of properties of tilings that are independent of dimension, and provides a method for transferring tilings from one dimension to another. Three well-known aperiodic tilings illustrate aspects of the correspondence: a tiling of Ammann, a Penrose tiling, and the monotiling of Taylor and Socolar-Taylor.