Deciding multiple tiling by polygons in polynomial time

TL;DR

This paper presents a polynomial time algorithm to determine if a symmetric convex polygon can tile the plane through multiple translations, by selecting vectors that form a discrete additive subgroup.

Contribution

It introduces a novel polynomial time algorithm for deciding multiple tiling by polygons, focusing on vector selection to form additive subgroups.

Findings

Algorithm successfully determines multiple tiling possibilities for symmetric convex polygons.

The method efficiently identifies vector sets that enable tiling.

Provides a polynomial time solution to a geometric tiling problem.

Abstract

Suppose $P$ is a symmetric convex polygon in the plane. We give a polynomial time algorithm that decides if $P$ can tile the plane by transations at some level (not necessarily at level one; this is multiple tiling). The main technical contribution is a polynomial time algorithm that selects, if this is possible, for each $j=1,2,\ldots,n$ one of two given vectors $e_j$ or $\tau_j$ so that the selection spans a discrete additive subgroup.