Ingeometry, apolyominoid (orminoid for short) is a set of equalsquares in3D space, joined edge to edge at 90- or 180-degree angles. The polyominoids include thepolyominoes, which are just the planar polyominoids. The surface of acube is an example of ahexominoid, or 6-cell polyominoid, and many otherpolycubes have polyominoids as their boundaries. Polyominoids appear to have been first proposed byRichard A. Epstein.[1]
90-degree connections are calledhard; 180-degree connections are calledsoft. 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 ashard if every junction includes a 90° connection,soft if every connection is 180°, andmixed otherwise, except in the unique case of the monominoid, which has no connections of either kind. The set of soft polyominoids is equal to the set ofpolyominoes.
As with otherpolyforms, two polyominoids that are mirror images may be distinguished.One-sided polyominoids distinguish mirror images;free polyominoids do not.
The table below enumerates free and one-sided polyominoids of up to 6 cells.
| Free | One-sided Total[3] | ||||
|---|---|---|---|---|---|
| Cells | Soft | Hard | Mixed | Total[4] | |
| 1 | see above | 1 | 1 | ||
| 2 | 1 | 1 | 0 | 2 | 2 |
| 3 | 2 | 5 | 2 | 9 | 11 |
| 4 | 5 | 16 | 33 | 54 | 80 |
| 5 | 12 | 89 | 347 | 448 | 780 |
| 6 | 35 | 526 | 4089 | 4650 | 8781 |
In general one can define ann,k-polyominoid as apolyform made by joiningk-dimensional hypercubes at 90° or 180° angles inn-dimensional space, where 1≤k≤n.
