Conditions for Discrete Equidecomposability of Polygons

TL;DR

This paper establishes a complete criterion and algorithm for determining when two rational polygons can be partitioned and reassembled into each other using lattice-preserving affine transformations.

Contribution

It extends previous work by providing a necessary and sufficient condition for rational finite discrete equidecomposability of polygons.

Findings

Developed a complete criterion for equidecomposability.

Created an algorithm to detect and construct equidecomposability relations.

Extended the invariant 'weight' to characterize equidecomposability.

Abstract

Two rational polygons $P$ and $Q$ are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice $\mathbb{Z} \times \mathbb{Z}$) from $P$ to $Q$. In [TW14], we developed an invariant for rational finite discrete equidecomposability known as weight. Here we extend this program with a necessary and sufficient condition for rational finite discrete equidecomposability. We close with an algorithm for detecting and constructing equidecomposability relations between rational polygons $P$ and $Q$.