A pentacube is a polycube composed of 5 cubes. There are 29 distinct three-dimensional pentacubes (Bouwkamp 1981). Of these, the 12 planar pentacubes (corresponding to solid pentominoes), are well known. Among the nonplanar pentacubes, there are five that have at least one plane of symmetry; each of them is its own mirror image. The remaining 12 pentacubes come in mirror image pairs.
The full family of 29 pentacubes, with mirror images distinguished, is also the piece set for a polycube wall. Its volume optimization counts enclosed free cubes rather than the cubes occupied by the pieces.
