Geodesic
High-dimensional holeyominoes
Greg Malen1 ; Fedor Manin2 ; Érika Roldán3
1Department of Mathematics, Union College
2Department of Mathematics, UCSB
3TUM and EPFL
The electronic journal of combinatorics, Tome 29 (2022) no. 3
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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.
DOI : 10.37236/10515
Classification : 05B50, 05A16, 05A20, 05D99
Mots-clés : polyominoes, toric polycube, Lee metric

Greg Malen  1   ; Fedor Manin  2   ; Érika Roldán  3

1 Department of Mathematics, Union College
2 Department of Mathematics, UCSB
3 TUM and EPFL 
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Greg Malen; Fedor Manin; Érika Roldán. High-dimensional holeyominoes. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/10515

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