Polycube
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Image:AGK-pentacube.png In recreational mathematics, a polycube is a polyform with a cube as the base form. Consequently they are the three-dimensional analogues of the planar polyominos. The Soma cube is an example of a packing problem based on polycubes.
Enumerating polycubes
As with polyominos, polycubes can be enumerated in two ways, depending on whether chiral solids are counted once (both reflections counted together) or twice (each reflection counted as distinct). For example, there is a single chiral tetracubes and 6 tetracubes with mirror symmetry, giving a count of 7 or 8 tetracubes respectively. Unlike with polyominos, the usual counting scheme is the latter one, reflecting the fact that in a physical set of polyominos, chiral forms can just be turned over to become their mirror images, something not possible for polycubes. For example, the Soma cube uses both forms of the chiral tetracube.
| n | Name of n-polycube | Number of n-polycubes (reflections counted as distinct) | Number of n-polycubes (reflections counted together) |
|---|---|---|---|
| 1 | 1 | 1 | |
| 2 | 1 | 1 | |
| 3 | tricube | 2 | 2 |
| 4 | tetracube | 8 | 7 |
| 5 | pentacube | 29 | 23 |
| 6 | hexacube | 166 | |
| 7 | heptacube | 1023 | |
| 8 | octocube | 6922 |