Editing Polyform

{{Short description|2D shape constructed by joining together identical basic polygons}}
[[Image:All 18 Pentominoes.svg|thumb|The 18 one-sided [[pentomino]]es: polyforms consisting of five squares.]]

In [[recreational mathematics]], a '''polyform''' is a [[plane (mathematics)|plane]] figure or solid compound constructed by joining together identical basic [[polygon]]s. The basic polygon is often (but not necessarily) a [[convex polygon|convex]] plane-filling polygon, such as a [[Square (geometry)|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 [[polyomino]]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:
#Two basic polygons may be joined only along a common edge, and must share the entirety of that edge.
#No two basic polygons may overlap.
#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.
#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]] can be joined along congruent faces. Joining [[cube (geometry)|cube]]s in this way produces the [[polycube]]s, and joining [[tetrahedron]]s 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 (polyhedron)|net]]; in the case of polyominoes, this results in [[polyominoid]]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 tile]]s 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-polyomino]]s, also known as polyplets or polykings.<ref>{{MathWorld|urlname=Polyplet|title=Polyplet}}</ref>

==Types and applications==
Polyforms are a rich source of problems, [[puzzle]]s and [[game]]s. 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.

{| class=wikitable
|+ Regular polygons
|-
!Sides
!colspan="2"|Basic polygon (monoform)
!width=170|Monohedral<BR>tessellation
!Polyform
!Applications
|-
!3
![[image:Monoiamond.png]]
|[[Triangle#Types of triangle|equilateral triangle]]
|[[File:Uniform triangular tiling 111111.png|80px]]<BR>[[Deltille]]
|[[Polyiamond]]s: moniamond, diamond, triamond, tetriamond, pentiamond, hexiamond
|[[Blokus Trigon]]
|-
!4
![[image:Monomino.png]]
|[[Square (geometry)|square]]
|[[File:Square tiling uniform coloring 1.svg|80px]]<BR>[[Square tiling|Quadrille]]
|[[Polyomino]]s: monomino, [[Domino (mathematics)|domino]], [[tromino]], [[tetromino]], [[pentomino]], [[hexomino]], [[heptomino]], [[octomino]], [[nonomino]], [[decomino]]
|[[Tetris]], [[Fillomino]], [[Tentai Show]], [[Ripple Effect (puzzle)]], [[LITS]], [[Nurikabe (puzzle)|Nurikabe]], [[Sudoku]], [[Blokus]]
|-
!6
![[image:Monohex.png]]
|[[Hexagon|regular hexagon]]
|[[File:Uniform tiling 63-t0.svg|80px]]<BR>[[Hextille]]
|[[polyhex (mathematics)|Polyhex]]es: monohex, dihex, trihex, tetrahex, pentahex, hexahex
|
|}

{| class=wikitable
|+ Other polyforms
|-
!Sides
!colspan="2"|Basic polygon (monoform)
!width=170|Monohedral<BR>tessellation
!Polyform
!Applications
|-
!1
![[image:Monostick.png]]
|[[line segment]]
|
|[[polystick]]
|[[Display device#Segment displays|Segment Displays]]
|-
!rowspan=2|3
![[image:Monodrafter.png]]
|[[Special right triangles#30–60–90 triangle|30°-60°-90° triangle]]
|[[File:1-uniform 3 dual.svg|80px]]<BR>[[Kisrhombille tiling|Kisrhombille]]
|[[polydrafter]]
|[[Eternity puzzle]]
|-
![[image:Monoabolo.png]]
|[[Special right triangles#45–45–90 triangle|right isosceles (45°-45°-90°) triangle]]
|[[File:1-uniform 2 dual.svg|80px]]<BR>[[Kisquadrille]]
|[[polyabolo]]
|[[Tangrams]]
|-
!4
![[Image:Monominoid.svg|60px]]
|[[rhombus]]
|[[File:Rhombic star tiling.svg|80px]]<BR>[[Rhombille]]
|[[polyrhomb]]
|
|-
!4
!
|Joined Half-Squares
|
|[[Polyare]]
|
|-
!12
!
|Joined Half-Cubes
|
|[[Polybe]]
|
|-
!5
![[Image:Pentagonal Cairo Snub Square Tile.svg|60px]]
|[[Cairo pentagonal tiling|Cairo Pentagon]]
|
|[[Polycairo]]
|
|-
!12
![[Image:Hexahedron.svg|60px]]
|[[Cube]]
|
|[[Polycube]]
|[[Soma cube]], [[Bedlam cube]], [[Diabolical cube]], [[Slothouber–Graatsma puzzle]], [[Conway puzzle]]
|-
!4
!
|Joined Half-Hexagons
|
|[[Polyhe]]
|
|-
!4
!
|[[Kite_(geometry)|60°-90°-90°-120° Kite]]
|
|[[Polykite]]
|
|-
!4
![[image:Monomino.png]]
|Square (Connected at Edges or Corners)
|
|[[Polyplet]]
|
|-
!3
!
|30°-30°-120° Isosceles Triangle
|
|[[Polypon]]
|
|-
!4
!
|[[Rectangle]]
|
|[[Polyrect]]
|
|}

== See also ==
*[[Polycube]]
*[[Polyomino]]
*[[Polyominoid]]

==References==
<references/>

==External links==
{{commons category|Polyforms}}
*{{MathWorld|urlname=Polyform|title=Polyform}}
*[http://www.recmath.org/PolyPages/index.htm ''The Poly Pages'' at RecMath.org], illustrations and information on many kinds of polyforms.

{{Polyforms}}

[[Category:Polyforms| ]]