Highdimensional holeyominoes
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 $f_d(n)/n \to (d1)/d$ 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 errorcorrecting codes and dynamical systems.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 DOI:
 10.48550/arXiv.2104.04558
 arXiv:
 arXiv:2104.04558
 Bibcode:
 2021arXiv210404558M
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Algebraic Topology;
 Mathematics  Geometric Topology;
 05A16;
 05A20;
 05B50;
 05D9
 EPrint:
 10 pages, 4 figures