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 ...

, a polyominoid (or minoid for short) is a set of equal

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...

s in 3D space, joined edge to edge at 90- or 180-degree angles. The polyominoids include the

polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in pop ...

es, which are just the planar polyominoids. The surface of a cube is an example of a ''hexominoid,'' or 6-cell polyominoid, and many other

polycube upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total A puzzle involving arranging nine L tricubes into a 3×3 cube A polycube is a solid figure formed by j ...

s have polyominoids as their boundaries. Polyominoids appear to have been first proposed by Richard A. Epstein.

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

es. As with other

polyform In recreational mathematics, a polyform is a plane figure or solid compound constructed by joining together identical basic polygons. The basic polygon is often (but not necessarily) a convex plane-filling polygon, such as a square or a triangle ...

s, 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

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 g ...

s are 2,1-polyominoids. *

Polyomino A polyomino is a plane geometric figure formed by joining one or more equal squares edge to edge. It is a polyform whose cells are squares. It may be regarded as a finite subset of the regular square tiling. Polyominoes have been used in pop ...

es are 2,2-polyominoids. *The polyforms described above are 3,2-polyominoids. *

Polycube upAll 8 one-sided tetracubes – if chirality is ignored, the bottom 2 in grey are considered the same, giving 7 free tetracubes in total A puzzle involving arranging nine L tricubes into a 3×3 cube A polycube is a solid figure formed by j ...

s are 3,3-polyominoids.

References

{{Polyforms Polyforms