Polyform

File:All 18 Pentominoes.svg

The 18 one-sided pentominoes: polyforms consisting of five squares.

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.

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, and must share the entirety of that edge.
2. No two basic polygons may overlap.
3. A polyform must be connected (that is, all one piece; see connected graph, connected space). 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").

These construction rules are not meant to be set in stone, but rather serve as general guidelines as to how polyforms may be constructed. Modifications of the first construction rule, for example, lead to different polyforms. Joining at a common vertex may lead to polykings, and being joined not by edge, but by the chess movement of the knight may lead to polyknights.

Generalizations

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-polyominoes, also known as polyplets or polykings.

Types and applications

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.

Regular polyforms

Sides	Basic polygon (monoform)		Monohedral tessellation	Polyform	Applications
3	File:Monoiamond.png	equilateral triangle	File:Uniform triangular tiling 111111.svg Deltille	Polyiamonds: moniamond, diamond, triamond, tetriamond, pentiamond, hexiamond	Blokus Trigon
4	File:Monomino.png	square	File:Square tiling uniform coloring 1.svg Quadrille	Polyominoes: monomino, domino, tromino, tetromino, pentomino, hexomino, heptomino, octomino, nonomino, decomino	Tetris, Fillomino, Tentai Show, Ripple Effect (puzzle), LITS, Nurikabe, Sudoku, Blokus
6	File:Monohex.png	regular hexagon	File:Uniform tiling 63-t0.svg Hextille	Polyhexes: monohex, dihex, trihex, tetrahex, pentahex, hexahex	Tantrix

Other low-dimensional polyforms

Sides	Basic polygon (monoform)		Monohedral tessellation	Polyform	Applications
1	File:Monostick.png	line segment (square)	-	Polysticks: monostick, distick, tristick, tetrastick, pentastick, hexastick	Segment Displays
		line segment (triangular)		Polytrigs
		line segment (hexagonal)		Polytwigs: monotwig, ditwig, tritwig, tetratwig, pentatwig, hexatwig
3	File:Monodrafter.png	30°-60°-90° triangle	File:1-uniform 3 dual.svg Kisrhombille	Polydrafters: monodrafter, didrafter, tridrafter, tetradrafter, pentadrafter, hexadrafter	Eternity puzzle
	File:Monoabolo.png	right isosceles (45°-45°-90°) triangle	File:1-uniform 2 dual.svg Kisquadrille	Polyaboloes: monabolo, diabolo, triabolo, tetrabolo, pentabolo, hexabolo, heptabolo, octabolo, enneabolo, decabolo	Tangram
		30°-30°-120° isosceles triangle	File:Tiling truncated 6 dual simple.svg Kisdeltille	Polypons: tripon, tetrapon
		golden triangle		Polyores
4	File:Monomino.png	square (connected at edges or corners)	File:Square tiling uniform coloring 1.svg Quadrille	Polykings: pentaking, hexaking, heptaking
		square (connected at edges, shifted by half)		Polyhops: dihop, trihop, tetrahop
		square (connected at edges in 3D space)		Polyominoids: monominoid
		square (representing path of a chess knight)		Polyknights: tetraknight, pentaknight, hexaknight	Knight in chess
		rectangle	File:Stacked bond.png Stacked bond	Polyrects: tetrarect, pentarect, hexarect, heptarect	Brickwork
		trapezoid		Polytraps: tritrap
	File:Monominoid.svg	rhombus	File:Rhombic star tiling.svg Rhombille	Polyrhombs
		60°-90°-90°-120° kite	File:Tiling small rhombi 3-6 dual simple.svg Tetrille	Polykites: trikite, tetrakite, pentakite, hexakite, heptakite
		half-squares		Polyares: triare, tetrare, pentare, hexare
		half-hexagons		Polyhes: monohe, dihe, trihe, tetrahe
5	File:Regular polygon 5 annotated.svg	regular pentagon	-	Polypents: monopent, dipent, tripent, tetrapent, pentapent, hexapent, heptapent
	File:Pentagonal Cairo Snub Square Tile.svg	Cairo pentagon	File:Equilateral Cairo tiling.svg 4-fold pentille	Polycairoes
		flaptile	File:1-uniform 8 dual.svg Iso(4-)pentille	Polyflaptiles: diflaptile, triflaptile, tetraflaptile
		120°-120°-120°-120°-60° pentagon	File:Tiling snub 3-6 left dual simple.svg 6-fold pentille	Polyflorets
6		Rombik		Polyrombiks
8	File:Regular polygon 8 annotated.svg	regular octagon (with squares)		Polyocts: dioct
-		quarter of circular arc		Polybends
	File:Circle-withsegments.svg	circle (with concave circles as bridges)		Polyrounds
		quarter of circle, and quarter-circle sector removed from a square		Polyarcs: monarc, diarc, triarc

High-dimensional polyforms

Edges	Basic polytope (monoform)		Monohedral honeycomb	Polyform	Applications
12	File:Cube-h.svg	cube	File:Cubic honeycomb.png Cubille	Polycubes: monocube, dicube, tricube, tetracube, pentacube, hexacube, heptacube, octacube	Soma cube, Bedlam cube, Diabolical cube, Snake cube, Slothouber–Graatsma puzzle, Conway puzzle, Herzberger Quader
		half-cubes		Polybes: monobe, dibe, tribe, hexabe
32	File:Hypercube.svg	tesseract	File:Tesseractic tetracomb.png Tesseractic honeycomb	Polytesseracts

See also

• Polycube
• Polyomino
• Polyominoid

External links

• The Poly Pages at RecMath.org, illustrations and information on many kinds of polyforms.
• Poly Puzzles, for polyform related puzzles.