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. More specific names have been given to polyforms resulting from specific basic polygons, as detailed in the table below. For example, a square basic polygon results in the well-known polyominoes.
The rules for joining the polygons together may vary, and must therefore be stated for each distinct type of polyform. Generally, however, the following rules apply:
Polyforms can also be considered in higher dimensions. In 3-dimensional space, basic polyhedra can be joined along congruent faces. Joining cubes in this way produces the polycubes, and joining tetrahedrons in this way produces the polytetrahedrons. 2-dimensional polyforms can also be folded out of the plane along their edges, in similar fashion to a net; in the case of polyominoes, this results in polyominoids.
One can allow more than one basic polygon. The possibilities are so numerous that the exercise seems pointless, unless extra requirements are brought in. For example, the Penrose tiles define extra rules for joining edges, resulting in interesting polyforms with a kind of pentagonal symmetry.
When the base form is a polygon that tiles the plane, rule 1 may be broken. For instance, squares may be joined orthogonally at vertices, as well as at edges, to form hinged/pseudo-polyominos, also known as polyplets or polykings. [1]
Polyforms are a rich source of problems, puzzles and games. The basic combinatorial problem is counting the number of different polyforms, given the basic polygon and the construction rules, as a function of n, the number of basic polygons in the polyform.
| Sides | Basic polygon (monoform) | Monohedral tessellation | Polyform | Applications | |
|---|---|---|---|---|---|
| 3 |  | equilateral triangle |  Deltille | Polyiamonds: moniamond, diamond, triamond, tetriamond, pentiamond, hexiamond | Blokus Trigon |
| 4 |  | square |  Quadrille | Polyominos: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, nonomino, decomino | Tetris, Fillomino, Tentai Show, Ripple Effect (puzzle), LITS, Nurikabe, Sudoku, Blokus |
| 6 |  | regular hexagon |  Hextille | Polyhexes: monohex, dihex, trihex, tetrahex, pentahex, hexahex | |
| Sides | Basic polygon (monoform) | Monohedral tessellation | Polyform | Applications | |
|---|---|---|---|---|---|
| 1 |  | line segment | polystick | Segment Displays | |
| 3 |  | 30°-60°-90° triangle |  Kisrhombille | polydrafter | Eternity puzzle |
 | right isosceles (45°-45°-90°) triangle |  Kisquadrille | polyabolo | Tangrams | |
| 4 |  | rhombus |  Rhombille | polyrhomb | |
| 4 | Joined Half-Squares | Polyare | |||
| 12 | Joined Half-Cubes | Polybe | |||
| 5 |  | Cairo Pentagon | Polycairo | ||
| 12 |  | Cube | Polycube | Soma cube, Bedlam cube, Diabolical cube, Slothouber–Graatsma puzzle, Conway puzzle | |
| 4 | Joined Half-Hexagons | Polyhe | |||
| 4 | 60°-90°-90°-120° Kite | Polykite | |||
| 4 | | Square (Connected at Edges or Corners) | Polyplet | ||
| 3 | 30°-30°-120° Isosceles Triangle | Polypon | |||
| 4 | Rectangle | Polyrect | |||
Derived from the Greek word for '5', and "domino", a pentomino is a polyomino of order 5; that is, a polygon in the plane made of 5 equal-sized squares connected edge to edge. When rotations and reflections are not considered to be distinct shapes, there are 12 different free pentominoes. When reflections are considered distinct, there are 18 one-sided pentominoes. When rotations are also considered distinct, there are 63 fixed pentominoes.
In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.
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.
A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.
A hexomino is a polyomino of order 6; that is, a polygon in the plane made of 6 equal-sized squares connected edge to edge. The name of this type of figure is formed with the prefix hex(a)-. When rotations and reflections are not considered to be distinct shapes, there are 35 different free hexominoes. When reflections are considered distinct, there are 60 one-sided hexominoes. When rotations are also considered distinct, there are 216 fixed hexominoes.
In recreational mathematics, a polystick is a polyform with a line segment 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 grid. A triangular polystick is a connected subset of a regular triangular grid. Polysticks are classified according to how many line segments they contain.
In recreational mathematics, a polydrafter is a polyform with a 30°–60°–90° right triangle as the base form. This triangle is also called a drafting triangle, hence the name. This triangle is also half of an equilateral triangle, and a polydrafter's cells must consist of halves of triangles in the triangular tiling of the plane; consequently, when two drafters share an edge that is the middle of their three edge lengths, they must be reflections rather than rotations of each other. Any contiguous subset of halves of triangles in this tiling is allowed, so unlike most polyforms, a polydrafter may have cells joined along unequal edges: a hypotenuse and a short leg.
In recreational mathematics, a polyabolo is a shape formed by gluing isosceles right triangles edge-to-edge, making a polyform with the isosceles right triangle as the base form. Polyaboloes were introduced by Martin Gardner in his June 1967 "Mathematical Games column" in Scientific American.
In recreational mathematics, a polyhex is a polyform with a regular hexagon as the base form, constructed by joining together 1 or more hexagons. Specific forms are named by their number of hexagons: monohex, dihex, trihex, tetrahex, etc. They were named by David Klarner who investigated them.
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.
A tromino or triomino is a polyomino of size 3, that is, a polygon in the plane made of three equal-sized squares connected edge-to-edge.
A heptomino is a polyomino of order 7; that is, a polygon in the plane made of 7 equal-sized squares connected edge to edge. The name of this type of figure is formed with the prefix hept(a)-. When rotations and reflections are not considered to be distinct shapes, there are 108 different free heptominoes. When reflections are considered distinct, there are 196 one-sided heptominoes. When rotations are also considered distinct, there are 760 fixed heptominoes.
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.
An octomino is a polyomino of order 8; that is, a polygon in the plane made of 8 equal-sized squares connected edge to edge. When rotations and reflections are not considered to be distinct shapes, there are 369 different free octominoes. When reflections are considered distinct, there are 704 one-sided octominoes. When rotations are also considered distinct, there are 2,725 fixed octominoes.
In geometry, a polyominoid 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.
In mathematics, a domino is a polyomino of order 2, that is, a polygon in the plane made of two equal-sized squares connected edge-to-edge. When rotations and reflections are not considered to be distinct shapes, there is only one free domino.
A pseudo-polyomino, also called a polyking, polyplet or hinged polyomino, is a plane geometric figure formed by joining one or more equal squares edge-to-edge or corner-to-corner at 90°. It is a polyform with square cells. The polyominoes are a subset of the polykings.
In the geometry of tessellations, a rep-tile or reptile is a shape that can be dissected into smaller copies of the same shape. The term was coined as a pun on animal reptiles by recreational mathematician Solomon W. Golomb and popularized by Martin Gardner in his "Mathematical Games" column in the May 1963 issue of Scientific American. In 2012 a generalization of rep-tiles called self-tiling tile sets was introduced by Lee Sallows in Mathematics Magazine.
Polyominoes: Puzzles, Patterns, Problems, and Packings is a mathematics book on polyominoes, the shapes formed by connecting some number of unit squares edge-to-edge. It was written by Solomon Golomb, and is "universally regarded as a classic in recreational mathematics". The Basic Library List Committee of the Mathematical Association of America has strongly recommended its inclusion in undergraduate mathematics libraries.
| Polyominoes | |
|---|---|
| Higher dimensions | |
| Others | |
| Games and puzzles | |
