Andrei Asinowski, Ronnie Barequet, Gill Barequet, and Günter Rote:

Proper n-cell polycubes in n−3 dimensions

  1. In: "Computing and Combinatorics". Proceedings of the 17th Annual International Computing and Combinatorics Conference (COCOON 2011), Dallas, Texas, August 2011. Editors: Bin Fu and Ding-Zhu Du. Lecture Notes in Computer Science 6842, Springer-Verlag, 2011, pp. 181–191. doi:10.1007/978-3-642-22685-4_16
  2. Journal of Integer Sequences 15 (2012), Article 12.8.4, 16 pages.


A d-dimensional polycube of size n is a connected set of n cubes in d dimensions, where connectivity is through (d−1)-dimensional faces. Enumeration of polycubes, and, in particular, specific types of polycubes, as well as computing the asymptotic growth rate of polycubes, is a popular problem in discrete geometry. This is also an important tool in statistical physics for computations related to percolation processes and branched polymers. In this paper we consider proper polycubes: A polycube is said to be proper in d dimensions if the convex hull of the centers of its cubes is d-dimensional. We prove a formula for the number of polycubes of size n that are proper in n−3 dimensions.

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Last update: October 3, 2012.