Improved Upper Bounds on the Growth Constants of Polyominoes and Polycubes

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 12118)


A d-dimensional polycube is a face-connected set of cells on \(\mathbb {Z}^d\). Let \(A_d(n)\) denote the number of d-dimensional polycubes (distinct up to translations) with n cubes, and \(\lambda _d\) denote their growth constant \( \lim _{n \rightarrow \infty }\frac{A_d(n{+}1)}{A_d(n)}\). We revisit and extend the method for the best known upper bound on \(A_2(n)\). Our contributions: We (1) prove that \(\lambda _2 \le 4.5252\); (2) prove that \(\lambda _d \le (2d-2)e+o(1)\) for \(d \ge 2\) (already improving significantly the upper bound on \(\lambda _3\) to 9.8073); and (3) implement an iterative process in 3D, improving further the upper bound on \(\lambda _3\) to 9.3835.


Klarner’s constant Square lattice Cubical lattice 


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Authors and Affiliations

  1. 1.Department of Computer ScienceThe Technion—Israel Institute of TechnologyHaifaIsrael
  2. 2.Department of Computer ScienceStanford UniversityStanfordUSA

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