In geometry, a polyominoid (or minoid for short) is a set of equal squares in 3D space, joined edge to edge at 90- or 180-degree angles. The polyominoids include the polyominoes, which are just the planar polyominoids. The surface of a cube is an example of a hexominoid, or 6-cell polyominoid. Polyominoids appear to have been first proposed by Richard A. Epstein.[1]
90-degree connections are called hard; 180-degree connections are called soft. This is because, in manufacturing a model of the polyominoid, a hard connection would be easier to realize than a soft one.[2] Polyominoids may be classified as hard, soft, and mixed according to how their edges are joined, except that the unique monominoid has no connections of either kind, which makes it both hard and soft by default. The number of soft polyominoids for each n equals the number of polyominoes for the same n.
As with other polyforms, two polyominoids that are mirror images may be distinguished. One-sided polyominoids distinguish mirror images; free polyominoids do not.
The table below enumerates soft, hard and mixed free polyominoids of up to 4 cells and the total number of free and one-sided polyominoids of up to 6 cells.
| Free | One-sided Total[3] |
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|---|---|---|---|---|---|
| Cells | Soft | Hard | Mixed | Total[4] | |
| 1 | 0 | 1 | 0 | 1 | 1 |
| 2 | 1 | 1 | 0 | 2 | 2 |
| 3 | 2 | 5 | 2 | 9 | 11 |
| 4 | 5 | 15 | 34 | 54[5] | 80 |
| 5 | ? | ? | ? | 448 | 780 |
| 6 | ? | ? | ? | 4650 | 8781 |
In general one can define an n,k-polyominoid as a polyform made by joining k-dimensional hypercubes at 90° or 180° angles in n-dimensional space, where 1≤k≤n.
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