Polycubes

Polycubes are the three-dimensional counterpart of polyominoes. They consist of n unit cubes connected face-to-face on a cubic lattice. The cubes occupy integer coordinates in Z^3, and the shape is determined by the set of occupied coordinates. Two polycubes are considered the same if one can be moved by rigid motions to align with the other; translations are always allowed.

Counting variants depends on symmetry. Fixed polycubes are distinct under translation only; rotating or reflecting a shape yields a new polycube. One-sided polycubes identify shapes that differ only by reflection as distinct, but consider translations and rotations equivalent. Free polycubes consider two shapes identical if they can be related by any isometry of three-dimensional space, including reflections.

Polycubes can be chiral; some shapes are mirror images that are not superimposable by any rotation, leading to different one-sided or fixed counts. Enumeration grows rapidly with n, and exact counts are known only for small n; researchers use backtracking and canonical forms to avoid duplicates. The set of all polycubes of size n is related to lattice animals in statistical mechanics.

Polycubes appear in tiling and packing problems, voxel-based modeling, 3D printing, and crystallography. They are studied alongside polyominoes and polyhedra as a combinatorial class of connected lattice animals, with numerous open questions about growth rates, symmetries, and enumeration algorithms.