Hexacubes

A hexacube is a three-dimensional solid composed of six identical cubes, where each cube shares at least one full face with another cube. These shapes are also known as hexacubes or hexa-cubes. They are a type of polycube, which are geometric shapes made by joining equal cubes face-to-face. The hexacube is the sixth order of polycube, following the monomino (1 cube), domino (2 cubes), tromino (3 cubes), tetromino (4 cubes), and pentomino (5 cubes).

There are 35 distinct free hexacubes, meaning that rotations and reflections are considered the same shape. If reflections are considered distinct, there are 45 one-sided hexacubes. The study of polycubes, including hexacubes, is a topic in recreational mathematics and has connections to combinatorics and tiling problems.

Hexacubes can be used in puzzles, where the goal is to assemble a set of hexacubes into a specific shape or container, or to tile a larger area. Their construction from unit cubes makes them amenable to digital representation and algorithmic exploration. The number of possible hexacubes grows rapidly with the order of the polycube.