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, and many other polycubes have polyominoids as their boundaries. 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 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.
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 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.
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.
A polyiamond is a polyform whose base form is an equilateral triangle. The word polyiamond is a back-formation from diamond, because this word is often used to describe the shape of a pair of equilateral triangles placed base to base, and the initial 'di-' looks like a Greek prefix meaning 'two-'. The name was suggested by recreational mathematics writer Thomas H. O'Beirne in New Scientist 1961 number 1, page 164.
90 (ninety) is the natural number preceded by 89 and followed by 91.
24 (twenty-four) is the natural number following 23 and preceding 25.
35 (thirty-five) is the natural number following 34 and preceding 36.
68 (sixty-eight) is the natural number following 67 and preceding 69. It is an even number.
220 is the natural number following 219 and preceding 221.
700 is the natural number following 699 and preceding 701.
116 is the natural number following 115 and preceding 117.
A polycube is a solid figure formed by joining one or more equal cubes face to face. Polycubes are the three-dimensional analogues of the planar polyominoes. The Soma cube, the Bedlam cube, the Diabolical cube, the Slothouber–Graatsma puzzle, and the Conway puzzle are examples of packing problems based on polycubes.
213 is the number following 212 and preceding 214.
196 is the natural number following 195 and preceding 197.
100,000 (one hundred thousand) is the natural number following 99,999 and preceding 100,001. In scientific notation, it is written as 105.
A nonomino is a polyomino of order 9, that is, a polygon in the plane made of 9 equal-sized squares connected edge-to-edge. The name of this type of figure is formed with the prefix non(a)-. When rotations and reflections are not considered to be distinct shapes, there are 1,285 different free nonominoes. When reflections are considered distinct, there are 2,500 one-sided nonominoes. When rotations are also considered distinct, there are 9,910 fixed nonominoes.
216 is the natural number following 215 and preceding 217. It is a cube, and is often called Plato's number, although it is not certain that this is the number intended by Plato.
186 is the natural number following 185 and preceding 187.
277 is the natural number following 276 and preceding 278.
205 is the natural number following 204 and preceding 206.
238 is the natural number following 237 and preceding 239.
Polytetrahedron is a term used for three distinct types of objects, all based on the tetrahedron:
| Polyominoes | |
|---|---|
| Higher dimensions | |
| Others | |
| Games and puzzles | |
