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arXiv · 2104.04558

High-dimensional holeyominoes

Greg Malen·Fedor Manin·Erika Roldan
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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.

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Greg Malen, Fedor Manin, Erika Roldan. 2021-04-09. High-dimensional holeyominoes. https://doi.org/10.37236/10515

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