Multi-tiling and equidecomposability of polytopes by lattice translates

TL;DR

This paper characterizes polytopes that multi-tile space via lattice translations and provides conditions for their equidecomposability, advancing understanding of geometric tiling and decomposition in higher dimensions.

Contribution

It introduces a characterization of polytopes that multi-tile space and establishes a criterion for their equidecomposability using lattice translations.

Findings

Characterization of polytopes that multi-tile space by lattice translations

Necessary and sufficient conditions for polytopes to be equidecomposable by lattice translations

Applicable to non-convex and disconnected polytopes in any dimension

Abstract

We characterize the polytopes in $\mathbb{R}^d$ (not necessarily convex or connected ones) which multi-tile the space by translations along a given lattice. We also give a necessary and sufficient condition for two polytopes in $\mathbb{R}^d$ to be equidecomposable by lattice translations.