Formulae and Growth Rates of High-Dimensional Polycubes

A d-dimensional polycube is a facet-connected set of cubes in d dimensions. Fixed polycubes are considered distinct if they differ in shape or orientation. A proper d-D polycube spans all d dimensions. In this paper we prove some formulae for fixed (proper and improper) polycubes, show that the growth-rate limit of the number of polycubes in d dimensions is 2 ed − o ( d ) , and estimate it at ( 2 d − 3 ) e + O ( 1 / d ) .