Editing Hexagon (section)

=== Symmetry ===
[[File:Hexagon reflections.svg|thumb|160px|The six lines of [[reflection symmetry|reflection]] of a regular hexagon, with Dih<sub>6</sub> or '''r12''' symmetry, order 12.]]
[[File:Regular hexagon symmetries.svg|thumb|400px|The dihedral symmetries are divided depending on whether they pass through vertices ('''d''' for diagonal) or edges ('''p''' for perpendiculars) Cyclic symmetries in the middle column are labeled as '''g''' for their central gyration orders. Full symmetry of the regular form is '''r12''' and no symmetry is labeled '''a1'''.]]

A regular hexagon has six [[rotational symmetries]] (''rotational symmetry of order six'') and six [[reflection symmetries]] (''six lines of symmetry''), making up the [[dihedral group]] D<sub>6</sub>.<ref>{{citation
 | last1 = Johnston | first1 = Bernard L.
 | last2 = Richman | first2 = Fred
 | year = 1997
 | publisher = CRC Press
 | title = Numbers and Symmetry: An Introduction to Algebra
 | url = https://books.google.com/books?id=koUfrlgsmUcC&pg=PA92
 | page = 92
| isbn = 978-0-8493-0301-2
 }}.</ref> There are 16 subgroups. There are 8 up to isomorphism: itself (D<sub>6</sub>), 2 dihedral: (D<sub>3,</sub> D<sub>2</sub>), 4 [[cyclic group|cyclic]]: (Z<sub>6</sub>, Z<sub>3</sub>, Z<sub>2</sub>, Z<sub>1</sub>) and the trivial (e)

These symmetries express nine distinct symmetries of a regular hexagon. [[John Horton Conway|John Conway]] labels these by a letter and group order.<ref>John H. Conway, Heidi Burgiel, [[Chaim Goodman-Strauss]], (2008) The Symmetries of Things, {{ISBN|978-1-56881-220-5}} (Chapter 20, Generalized Schaefli symbols, Types of symmetry of a polygon pp. 275–278)</ref> '''r12''' is full symmetry, and '''a1''' is no symmetry. '''p6''', an [[isogonal figure|isogonal]] hexagon constructed by three mirrors can alternate long and short edges, and '''d6''', an [[isotoxal figure|isotoxal]] hexagon constructed with equal edge lengths, but vertices alternating two different internal angles. These two forms are [[dual polygon|duals]] of each other and have half the symmetry order of the regular hexagon. The '''i4''' forms are regular hexagons flattened or stretched along one symmetry direction. It can be seen as an [[Elongation (geometry)|elongated]] [[rhombus]], while '''d2''' and '''p2''' can be seen as horizontally and vertically elongated [[Kite (geometry)|kites]]. '''g2''' hexagons, with opposite sides parallel are also called hexagonal [[parallelogon]]s.

Each subgroup symmetry allows one or more degrees of freedom for irregular forms. Only the '''g6''' subgroup has no degrees of freedom but can be seen as [[directed edge]]s.

Hexagons of symmetry '''g2''', '''i4''', and '''r12''', as [[parallelogon]]s can tessellate the Euclidean plane by translation. Other [[Hexagonal tiling#Topologically equivalent tilings|hexagon shapes can tile the plane]] with different orientations.
{| class=wikitable
!''p''6''m'' (*632)
!''cmm'' (2*22)
!''p''2 (2222)
!''p''31''m'' (3*3)
!colspan=2|''pmg'' (22*)
!''pg'' (××)
|-
![[File:Isohedral_tiling_p6-13.svg|120px]]<BR>[[hexagonal tiling|r12]]
![[File:Isohedral_tiling_p6-12.png|120px]]<BR>i4
![[File:Isohedral_tiling_p6-7.svg|120px]]<BR>g2
![[File:Isohedral tiling p6-11.png|120px]]<BR>d2
![[File:Isohedral tiling p6-10.png|120px]]<BR>d2
![[File:Isohedral tiling p6-9.svg|120px]]<BR>p2
![[File:Isohedral tiling p6-1.png|120px]]<BR>a1
|- valign=top al
!Dih<sub>6</sub>
!Dih<sub>2</sub>
!Z<sub>2</sub>
!colspan=3|Dih<sub>1</sub>
!Z<sub>1</sub>
|}

{| class="wikitable skin-invert-image" align=right style="text-align:center;"
|-
| [[File:Root system A2.svg|120px]]<BR>A2 group roots<BR>{{Dynkin|node_n1|3|node_n2}}
| [[File:Root system G2.svg|120px]]<BR>G2 group roots<BR>{{Dynkin2|nodeg_n1|6a|node_n2}}
|}
The 6 roots of the [[simple Lie group]] [[Dynkin diagram#Example: A2|A2]], represented by a [[Dynkin diagram]] {{Dynkin|node_n1|3|node_n2}}, are in a regular hexagonal pattern. The two simple roots have a 120° angle between them.

The 12 roots of the [[Exceptional Lie group#Exceptional cases|Exceptional Lie group]] [[G2 (mathematics)|G2]], represented by a [[Dynkin diagram]] {{Dynkin2|nodeg_n1|6a|node_n2}} are also in a hexagonal pattern. The two simple roots of two lengths have a 150° angle between them.