View source for Polyiamond

{{short description|Polyform whose base form is an equilateral triangle}}
{{redirect|Triamond|the crystal structure|Laves graph}}
{{Commons category|Polyiamonds}}
A '''polyiamond''' (also '''polyamond''' or simply '''iamond''', or sometimes '''triangular polyomino'''<ref>{{cite web |url=https://oeis.org/A000577 |title=A000577 |last=Sloane |first=N.J.A. |date=July 9, 2021 |website=OEIS |publisher=The OEIS Foundation Inc. |access-date=July 9, 2021 |quote="triangular polyominoes (or triangular polyforms, or polyiamonds)"}}</ref>) 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 language|Greek]] prefix meaning 'two-' (though ''diamond'' actually derives from Greek '' ἀδάμας'' - also the basis for the word "adamant"). The name was suggested by recreational mathematics writer Thomas H. O'Beirne in ''New Scientist'' 1961 number 1, page 164.

==Counting==
The basic [[combinatorics|combinatorial question]] is, How many different polyiamonds exist with a given number of cells? Like [[polyomino]]es, polyiamonds may be either free or one-sided. Free polyiamonds are invariant under reflection as well as translation and rotation. One-sided polyiamonds distinguish reflections.

The number of free ''n''-iamonds for ''n'' = 1, 2, 3, ... is:

:1, 1, 1, 3, 4, 12, 24, 66, 160, ... {{OEIS|id=A000577}}.

The number of free polyiamonds with holes is given by {{OEIS2C|id=A070764}}; the number of free polyiamonds without holes is given by {{OEIS2C|id=A070765}}; the number of fixed polyiamonds is given by {{OEIS2C|id=A001420}}; the number of one-sided polyiamonds is given by {{OEIS2C|id=A006534}}.

{|class=wikitable
!Name
!width=75|Number of forms
!Forms
|-
|Moniamond
|align=center|1
|
{|style="background:transparent"
|-
|align=left style="border:0px"| [[image:Polyiamond-1-1.svg|50px]]
|align=left style="border:0px"|
|-
|}
|-
|Diamond
|align=center|1
|
{|style="background:transparent"
|-
|align=left style="border:0px"| [[image:Polyiamond-2-1.svg|50px]]
|align=left style="border:0px"|
|-
|}
|-
|Triamond
|align=center|1
|
{|style="background:transparent"
|-
|align=left style="border:0px"| [[image:Polyiamond-3-1.svg|50px]]
|align=left style="border:0px"|
|-
|}
|-
|Tetriamond
|align=center|3
|
{|style="background:transparent"
|-
|align=left style="border:0px"| [[image:Polyiamond-4-2.svg|50px]]
|align=left style="border:0px"| [[image:Polyiamond-4-1.svg|50px]]
|align=left style="border:0px"| [[image:Polyiamond-4-3.svg|50px]]
|-
|}
|-
|Pentiamond
|align=center|4
|
{|style="background:transparent"
|-
|align=left style="border:0px"| [[image:Polyiamond-5-1.svg|50px]]
|align=left style="border:0px"| [[image:Polyiamond-5-2.svg|50px]]
|align=left style="border:0px"| [[image:Polyiamond-5-3.svg|50px]]
|align=left style="border:0px"| [[image:Polyiamond-5-4.svg|50px]]
|-
|}
|-
|Hexiamond
|align=center|12
|
{|style="background:transparent"
|-
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-1.svg|50px]]|bar}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-2.svg|50px]]|crook}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-3.svg|50px]]|crown}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-4.svg|50px]]|sphinx}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-5.svg|50px]]|snake}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-6.svg|50px]]|yacht}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-7.svg|50px]]|bat}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-8.svg|50px]]|pistol}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-9.svg|50px]]|lobster}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-10.svg|50px]]|shoe}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-11.svg|50px]]|hexagon}}
|align=left style="border:0px"| {{Tooltip|[[image:Polyiamond-6-12.svg|50px]]|butterfly}}
|-
|}
|-
|}
Some authors also call the diamond ([[rhombus]] with a 60° angle) a ''calisson'' after the [[calisson|French sweet]] of similar shape.<ref>{{Cite book|url=https://books.google.com/books?id=2F_0DwAAQBAJ&pg=PA28|title = A Mathematical Space Odyssey: Solid Geometry in the 21st Century|isbn = 9781614442165|last1 = Alsina|first1 = Claudi|last2 = Nelsen|first2 = Roger B.|date = 31 December 2015| publisher=American Mathematical Soc. }}</ref><ref>{{Cite journal|url=http://jstor.org/stable/2325150|jstor = 2325150|last1 = David|first1 = Guy|last2 = Tomei|first2 = Carlos|title = The Problem of the Calissons|journal = The American Mathematical Monthly|year = 1989|volume = 96|issue = 5|pages = 429–431|doi = 10.1080/00029890.1989.11972212}}</ref>

==Symmetries==<!-- This section is linked from [[Symmetry]] -->
Possible [[symmetry|symmetries]] are mirror symmetry, 2-, 3-, and 6-fold rotational symmetry, and each combined with mirror symmetry.

2-fold rotational symmetry with and without mirror symmetry requires at least 2 and 4 triangles, respectively. 6-fold rotational symmetry with and without mirror symmetry requires at least 6 and 18 triangles, respectively. Asymmetry requires at least 5 triangles. 3-fold rotational symmetry without mirror symmetry requires at least 7 triangles.

In the case of only mirror symmetry we can distinguish having the symmetry axis aligned with the grid or rotated 30° (requires at least 4 and 3 triangles, respectively); ditto for 3-fold rotational symmetry, combined with mirror symmetry (requires at least 18 and 1 triangles, respectively).

[[File:Polyiamond Symmetries.svg|Polyiamond Symmetries]]

==Generalizations==
Like [[polyomino]]es, but unlike [[polyhex (mathematics)|polyhex]]es, polyiamonds have three-[[dimension]]al counterparts, formed by aggregating [[tetrahedron|tetrahedra]]. However, [[polytetrahedron|polytetrahedra]] do not tile 3-space in the way polyiamonds can tile 2-space.

==Tessellations==
Every polyiamond of order 8 or less tiles the plane, except for the V-heptiamond.<ref>[http://www.mathpuzzle.com/Tessel.htm "All of the polyiamonds of order eight or less, with the exception of one of the heptiamonds will tessellate the plane. The exception is the V-shaped heptiamond. Gardner (6th book p.248) posed the problem of identifying this heptiamond and reproduced an impossibilty proof of Gregory. However, in combination with other heptiamonds or other polyiamonds, tesselations using this V-shaped heptiamond can be achieved."]</ref>

==Correspondence with polyhexes==
[[File:Mondtohex.svg|thumb|Pentiamond with corresponding pentahex superimposed.]]
Every polyiamond corresponds to a [[polyhex (mathematics)|polyhex]], as illustrated at right. Conversely, every polyhex is also a polyiamond, because each hexagonal cell of a polyhex is the union of six adjacent equilateral triangles. Neither correspondence is one-to-one.

==In popular culture==
The set of 22 polyiamonds, from order 1 up to order 6, constitutes the shape of the playing pieces in the board game [[Blokus Trigon]], where players attempt to tile a plane with as many polyiamonds as possible, subject to the game rules.

== See also ==
*[[Triangular tiling]]
*[[Rhombille tiling]]
*[[Sphinx tiling]]

==External links==
*{{MathWorld|title=Polyiamond|urlname=Polyiamond}}
*[http://www.recmath.org/PolyPages/PolyPages/Polyiamonds.htm Polyiamonds] at [http://www.recmath.org/PolyPages/index.htm The Poly Pages]. Polyiamond tilings.
*[http://home.arcor.de/mdoege/verhext/ VERHEXT] &mdash; a 1960s puzzle game by Heinz Haber based on hexiamonds ({{Webarchive |url=https://web.archive.org/web/20160303224119/http://home.arcor.de/mdoege/verhext/|date=March 3, 2016}})

==References==
{{reflist}}

{{Polyforms}}

[[Category:Polyforms]]