Parallelogram tilings, Worms and Finite Orientations

TL;DR

This paper investigates the properties of plane tilings with parallelograms, proving that finite-shape tilings only use finitely many orientations of tiles, revealing structural constraints of such tilings.

Contribution

It establishes that parallelogram tilings with finitely many shapes have only finitely many tile orientations, a new insight into their geometric structure.

Findings

All tiles in such tilings occur in finitely many orientations.

The result applies to tilings with a finite set of parallelogram shapes.

Provides a foundational understanding of the orientation limitations in parallelogram tilings.

Abstract

This paper studies properties of tilings of the plane by parallelograms. In particular it is established that in parallelogram tilings using a finite number of shapes all tiles occur in only finitely many orientations.