In recreational mathematics

Recreational mathematics

Recreational mathematics is an umbrella term, referring to mathematical puzzles and mathematical games.Not all problems in this field require a knowledge of advanced mathematics, and thus, recreational mathematics often attracts the curiosity of non-mathematicians, and inspires their further study...

, a polyform is a plane

Plane (mathematics)

In mathematics, a plane is a flat, two-dimensional surface. A plane is the two dimensional analogue of a point , a line and a space...

 figure constructed by joining together identical basic polygon

Polygon

In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...

s. The basic polygon is often (but not necessarily) a convex

Convex polygon

In geometry, a polygon can be either convex or concave .- Convex polygons :A convex polygon is a simple polygon whose interior is a convex set...

 plane-filling polygon, such as a square

Square (geometry)

In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...

 or a triangle

Triangle

A triangle is one of the basic shapes of geometry: a polygon with three corners or vertices and three sides or edges which are line segments. A triangle with vertices A, B, and C is denoted ....

. 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 polyomino

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 with a connected interior....

es.

Construction rules

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:

1. Two basic polygons may be joined only along a common edge.
2. No two basic polygons may overlap.
3. A polyform must be connected (that is, all one piece; see connected graph, connected space
   Connected space
   In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint nonempty open subsets. Connectedness is one of the principal topological properties that is used to distinguish topological spaces...
   
   ). Configurations of disconnected basic polygons do not qualify as polyforms.
4. The mirror image of an asymmetric polyform is not considered a distinct polyform (polyforms are "double sided").

Generalizations

Polyforms can also be considered in higher dimensions. In 3-dimensional space, basic polyhedra

Polyhedron

In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...

 can be joined along congruent faces. Joining cubes in this way produces the polycube

Polycube

thumb|200px|right|The seven free tetracubesthumb|200px|right|A [[Chirality |chiral]] pentacubethumb|200px|right|Puzzle with a unique solution...

s.

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 polyplets or polykings.

Types and applications

Polyforms are a rich source of problems, puzzle

Puzzle

A puzzle is a problem or enigma that tests the ingenuity of the solver. In a basic puzzle, one is intended to put together pieces in a logical way in order to come up with the desired solution...

s and game

Game

A game is structured playing, usually undertaken for enjoyment and sometimes used as an educational tool. Games are distinct from work, which is usually carried out for remuneration, and from art, which is more often an expression of aesthetic or ideological elements...

s. The basic combinatorial

Combinatorics

Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...

 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. Well-known puzzles include the pentomino puzzle

Pentomino

A pentomino is a polyomino composed of five congruent squares, connected along their edges ....

 and the Soma cube

Soma cube

The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven pieces made out of unit cubes must be assembled into a 3x3x3 cube...

.

Basic polygon (monoform)		Polyform	Applications
	line segment Line segment In geometry, a line segment is a part of a line that is bounded by two end points, and contains every point on the line between its end points. Examples of line segments include the sides of a triangle or square. More generally, when the end points are both vertices of a polygon, the line segment...	polystick Polystick In recreational mathematics, a polystick is a polyform with a line segment as the basic shape. Polysticks result when identical line segments are joined together end-to-end at 0° or 90° angles....
	square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...	polyomino 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 with a connected interior....	pentomino puzzle Pentomino A pentomino is a polyomino composed of five congruent squares, connected along their edges .... , Lonpos puzzle Lonpos *Lonpos is a logic puzzle game in which the player must fit a fixed number of beaded game pieces of various shapes, to fill a set area such as a rectangle or pyramid. The shapes used in Lonpos puzzles are similar to the shapes known as pentominoes... , Fillomino Fillomino Fillomino is a type of logic puzzle published by Nikoli. Other published titles for the puzzle include Allied Occupation. As of 2005, three books consisting entirely of Fillomino puzzles have been published by Nikoli.-Rules:... , Tentai Show, Ripple Effect (puzzle), LITS LITS LITS, formerly known as Nuruomino , is a binary determination puzzle published by Nikoli.-Rules:LITS is played on a rectangular grid, typically 10×10; the grid is divided into polyominoes, none of which have fewer than four cells... , Nurikabe Nurikabe Nurikabe is a binary determination puzzle named for an invisible wall in Japanese folklore that blocks roads and delays foot travel... , Sudoku Sudoku is a logic-based, combinatorial number-placement puzzle. The objective is to fill a 9×9 grid with digits so that each column, each row, and each of the nine 3×3 sub-grids that compose the grid contains all of the digits from 1 to 9...
	equilateral triangle	polyiamond Polyiamond 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-" looked like a Greek prefix meaning...
	30°-60°-90° triangle	polydrafter Polydrafter In recreational mathematics, a polydrafter is a polyform with a triangle as the base form. The triangle has angles of 30°, 60° and 90°, like a set square—hence the name. The polydrafter was invented by Christopher Monckton, whose Eternity Puzzle was composed of 209 hexadrafters.The term...	Eternity puzzle Eternity puzzle Eternity is a tiling puzzle created by Christopher Monckton and launched by the Ertl Company in June 1999. Consisting of 209 pieces, it was marketed as being practically unsolveable, with a £1 million prize on offer for whoever could solve it within four years. The prize was paid out in October... , Tentai Show
	right isosceles (45°-45°-90°) triangle	polyabolo Polyabolo In recreational mathematics, a polyabolo is a polyform with an isosceles right triangle as the base form.The name is a back formation from 'diabolo' although the shape formed by joining two triangles at just one vertex is not a proper polyabolo...
	regular hexagon	polyhex Polyhex (mathematics) In recreational mathematics, a polyhex is a polyform with a regular hexagon as the base form.As with polyominoes, polyhexes may be enumerated as free polyhexes , fixed polyhexes and one-sided polyhexes In recreational mathematics, a polyhex is a polyform with a regular hexagon (or 'hex' for...
	cube 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. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...	polycube Polycube thumb|200px|right|The seven free tetracubesthumb|200px|right|A [[Chirality |chiral]] pentacubethumb|200px|right|Puzzle with a unique solution...	Soma cube Soma cube The Soma cube is a solid dissection puzzle invented by Piet Hein in 1933 during a lecture on quantum mechanics conducted by Werner Heisenberg. Seven pieces made out of unit cubes must be assembled into a 3x3x3 cube...
	square Square (geometry) In geometry, a square is a regular quadrilateral. This means that it has four equal sides and four equal angles...  (in three dimensions)	polyominoid Polyominoid 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. Polyominoids appear to...

External links

• The Poly Pages at RecMath.org, illustrations and information on many kinds of polyforms.