Editing Polyiamond (section)

==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>