Slopes of Tilings

TL;DR

This paper investigates the slopes of periodic tilings, characterizing which slopes are possible and proving they correspond exactly to recursively enumerable sets of rationals.

Contribution

It provides a complete characterization of achievable slopes in tilings, showing they match recursively enumerable sets of rationals, a novel theoretical result.

Findings

Slopes of tilings are exactly the recursively enumerable sets of rationals.

The paper characterizes the set of slopes achievable with periodic tilings.

It establishes a correspondence between tiling slopes and computability theory.

Abstract

We study here slopes of periodicity of tilings. A tiling is of slope if it is periodic along direction but has no other direction of periodicity. We characterize in this paper the set of slopes we can achieve with tilings, and prove they coincide with recursively enumerable sets of rationals.