High-dimensional holeyominoes

Abstract

What is the maximum number of holes enclosed by a $d$-dimensional polycube built of $n$ tiles? Represent this number by $f_d\left(n\right)$. Recent results show that $f_2\left(n\right)/n$ converges to ¹⁄₂. We prove that for all $d\ge 2$ we have $f_d\left(n\right)/n\to (d-1)/d$ as $n$ goes to infinity. We also construct polycubes 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.