geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, 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 In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

is an example of a ''hexominoid,'' or 6-cell polyominoid, and many other polycubes have polyominoids as their boundaries. Polyominoids appear to have been first proposed by

Richard A. Epstein Richard Allen Epstein (born April 17, 1943) is an American legal scholar known for his writings on torts, contracts, property rights, law and economics, classical liberalism, and libertarianism. He is the Laurence A. Tisch Professor of Law at ...

Classification

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.The Polyominoids
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The Polyominoids
Polyominoids may be classified as ''hard'' if every junction includes a 90° connection, ''soft'' if every connection is 180°, and ''mixed'' 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 of polyominoes. As with other polyforms, two polyominoids that are mirror images may be distinguished. ''One-sided'' polyominoids distinguish mirror images; ''free'' polyominoids do not.

Enumeration

The table below enumerates free and one-sided polyominoids of up to 6 cells.

Generalization to higher dimensions

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''. *

Polystick In recreational mathematics, a polystick (or polyedge) is a polyform with a line segment (a 'stick') as the basic shape. A polystick is a connected set of segments in a regular grid. A square polystick is a connected subset of a regular square gr ...

s are 2,1-polyominoids. * Polyominoes are 2,2-polyominoids. *The polyforms described above are 3,2-polyominoids. * Polycubes are 3,3-polyominoids.

References

{{Polyforms Polyforms