High-dimensional holeyominoes

TL;DR

This paper investigates the maximum number of holes in high-dimensional polyominoes, establishing asymptotic ratios for the number of holes per tile and constructing optimal polyominoes using concepts from coding theory and dynamical systems.

Contribution

It generalizes previous two-dimensional results to all higher dimensions, proving the asymptotic ratio of holes per tile and constructing polyominoes with maximal holes.

Findings

Asymptotic ratio of holes per tile approaches (d-1)/d in d dimensions

Constructed polyominoes in tori with maximal holes per tile

Used error-correcting codes and dynamical systems in proofs

Abstract

What is the maximum number of holes enclosed by a $d$-dimensional polyomino built of $n$ tiles? Represent this number by $f_d(n)$. Recent results show that $f_2(n)/n$ converges to $1/2$. We prove that for all $d \geq 2$ we have $f_d(n)/n \to (d-1)/d$ as $n$ goes to infinity. We also construct polyominoes in $d$-dimensional tori with the maximal possible number of holes per tile. In our proofs, we use metaphors from error-correcting codes and dynamical systems.