Editing Polyomino (section)

===Symmetries of polyominoes===
The [[dihedral group]] ''D''<sub>4</sub> is the [[group (mathematics)|group]] of [[symmetries]] ([[symmetry group]]) of a square. This group contains four rotations and four reflections. It is generated by alternating reflections about the ''x''-axis and about a diagonal. One free polyomino corresponds to at most 8 fixed polyominoes, which are its images under the symmetries of ''D''<sub>4</sub>. However, those images are not necessarily distinct: the more symmetry a free polyomino has, the fewer distinct fixed counterparts it has. Therefore, a free polyomino that is invariant under some or all non-trivial symmetries of ''D''<sub>4</sub> may correspond to only 4, 2 or 1 fixed polyominoes. Mathematically, free polyominoes are [[equivalence class]]es of fixed polyominoes under the group ''D''<sub>4</sub>.

Polyominoes have the following possible symmetries;<ref name="Redelmeier, section 3">Redelmeier, section 3</ref> the least number of squares needed in a polyomino with that symmetry is given in each case:
*8 fixed polyominoes for each free polyomino:
**no symmetry (4)
*4 fixed polyominoes for each free polyomino:
**mirror symmetry with respect to one of the grid line directions (4)
**mirror symmetry with respect to a diagonal line (3)
**2-fold rotational symmetry: ''C''<sub>2</sub> (4)
*2 fixed polyominoes for each free polyomino:
**symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: ''D''<sub>2</sub> (2) (also known as the [[Klein four-group]])
**symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: ''D''<sub>2</sub> (7)
**4-fold rotational symmetry: ''C''<sub>4</sub> (8)
*1 fixed polyomino for each free polyomino:
**all symmetry of the square: ''D''<sub>4</sub> (1).

In the same way, the number of one-sided polyominoes depends on polyomino symmetry as follows:
*2 one-sided polyominoes for each free polyomino:
**no symmetry
**2-fold rotational symmetry: ''C''<sub>2</sub>
**4-fold rotational symmetry: ''C''<sub>4</sub>
*1 one-sided polyomino for each free polyomino:
**all symmetry of the square: ''D''<sub>4</sub>
**mirror symmetry with respect to one of the grid line directions
**mirror symmetry with respect to a diagonal line
**symmetry with respect to both grid line directions, and hence also 2-fold rotational symmetry: ''D''<sub>2</sub>
**symmetry with respect to both diagonal directions, and hence also 2-fold rotational symmetry: ''D''<sub>2</sub>.

The following table shows the numbers of polyominoes with ''n'' squares, sorted by symmetry groups. 
{|class=wikitable
!''n''
!none
!mirror<br/>90°
!mirror<br/>45°
!''C''<sub>2</sub>
!''D''<sub>2</sub><br/>90°
!''D''<sub>2</sub><br/>45°
!''C''<sub>4</sub>
!''D''<sub>4</sub>
|- align=right
|1 ||0 ||0 ||0 ||0 ||0 ||0 ||0 ||1
|- align=right
|2 ||0 ||0 ||0 ||0 ||1 ||0 ||0 ||0
|- align=right
|3 ||0 ||0 ||1 ||0 ||1 ||0 ||0 ||0
|- align=right
|4 ||1 ||1 ||0 ||1 ||1 ||0 ||0 ||1
|- align=right
|5 ||5 ||2 ||2 ||1 ||1 ||0 ||0 ||1
|- align=right
|6 ||20 ||6 ||2 ||5 ||2 ||0 ||0 ||0
|- align=right
|7 ||84 ||9 ||7 ||4 ||3 ||1 ||0 ||0
|- align=right
|8 ||316 ||23 ||5 ||18 ||4 ||1 ||1 ||1
|- align=right
|9 ||1,196 ||38 ||26 ||19 ||4 ||0 ||0 ||2
|- align=right
|10 ||4,461 ||90 ||22 ||73 ||8 ||1 ||0 ||0
|- align=right
|11 ||16,750 ||147 ||91 ||73 ||10 ||2 ||0 ||0
|- align=right
|12 ||62,878 ||341 ||79 ||278 ||15 ||3 ||3 ||3
|-
| [[OEIS]]&nbsp;sequence
|{{OEIS link|id=A006749}}
|{{OEIS link|id=A006746}}
|{{OEIS link|id=A006748}}
|{{OEIS link|id=A006747}}
|{{OEIS link|id=A056877}}
|{{OEIS link|id=A056878}}
|{{OEIS link|id=A144553}}
|{{OEIS link|id=A142886}}
|}

<ref>{{cite journal|doi=10.1016/0012-365X(81)90237-5 |doi-access=free |title=Counting polyominoes: Yet another attack |year=1981 |last1=Redelmeier |first1=D.Hugh |journal=Discrete Mathematics |volume=36 |issue=2 |pages=191–203 }}</ref>