On multiple translative tiling in the plane

TL;DR

This paper proves that any multiple translative tiling of the plane necessarily corresponds to a multiple lattice tiling, establishing a fundamental geometric relationship.

Contribution

It demonstrates that all multiple translative tiles in the plane are inherently multiple lattice tiles, clarifying their structural nature.

Findings

Multiple translative tiles are equivalent to multiple lattice tiles in the plane.

The result simplifies the classification of tilings in the plane.

Provides a foundational understanding of translative tiling structures.

Abstract

This paper shows that a multiple translative tile in the plane must be a multiple lattice tile.