Editing Hexagon (section)

==Regular hexagon==
A regular hexagon is defined as a hexagon that is both [[equilateral polygon|equilateral]] and [[equiangular polygon|equiangular]]. In other words, a hexagon is said to be regular if the edges are all equal in length, and each of its [[internal angle]] is equal to 120°. The [[Schläfli symbol]] denotes this polygon as <math> \{6\} </math>.<ref>{{citation
 | title = Polyhedron Models
 | first = Magnus J. | last = Wenninger
 | publisher = Cambridge University Press
 | year = 1974
 | page = 9
 | isbn = 9780521098595
 | url = https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9
 | access-date = 2015-11-06
 | archiveurl = https://web.archive.org/web/20160102075753/https://books.google.com/books?id=N8lX2T-4njIC&pg=PA9
 | archive-date = 2016-01-02
 | url-status = live
}}.</ref> However, the regular hexagon can also be considered as the [[Truncation (geometry)|cutting off the vertices]] of an [[equilateral triangle]], which can also be denoted as <math> \mathrm{t}\{3\} </math>.

A regular hexagon is [[bicentric polygon|bicentric]], meaning that it is both [[cyclic polygon|cyclic]] (has a circumscribed circle) and [[tangential polygon|tangential]] (has an inscribed circle). The common length of the sides equals the radius of the [[circumscribed circle]] or [[circumcircle]], which equals <math>\tfrac{2}{\sqrt{3}}</math> times the [[apothem]] (radius of the [[inscribed figure|inscribed circle]]).

=== Measurement ===
The longest diagonals of a regular hexagon, connecting diametrically opposite vertices, are twice the length of one side. From this it can be seen that a [[triangle]] with a vertex at the center of the regular hexagon and sharing one side with the hexagon is [[equilateral triangle|equilateral]], and that the regular hexagon can be partitioned into six equilateral triangles.

[[Image:Regular hexagon 1.svg|thumb|''R'' = [[Circumradius]]; ''r'' = [[Inradius]]; ''t'' = side length]]

The maximal [[diameter#Polygons|diameter]] (which corresponds to the long [[diagonal]] of the hexagon), ''D'', is twice the maximal radius or [[circumradius]], ''R'', which equals the side length, ''t''. The minimal diameter or the diameter of the [[inscribed]] circle (separation of parallel sides, flat-to-flat distance, short diagonal or height when resting on a flat base), ''d'', is twice the minimal radius or [[inradius]], ''r''. The maxima and minima are related by the same factor:
:<math>\frac{1}{2}d = r = \cos(30^\circ) R = \frac{\sqrt{3}}{2} R = \frac{\sqrt{3}}{2} t</math> &nbsp; and, similarly, <math>d = \frac{\sqrt{3}}{2} D.</math>

The area of a regular hexagon
:<math>\begin{align}
  A &= \frac{3\sqrt{3}}{2}R^2 = 3Rr           = 2\sqrt{3} r^2 \\[3pt]
    &= \frac{3\sqrt{3}}{8}D^2 = \frac{3}{4}Dd = \frac{\sqrt{3}}{2} d^2 \\[3pt]
    &\approx 2.598 R^2 \approx 3.464 r^2\\
    &\approx 0.6495 D^2 \approx 0.866 d^2.
\end{align}</math>

For any regular [[polygon]], the area can also be expressed in terms of the [[apothem]] ''a'' and the perimeter ''p''. For the regular hexagon these are given by ''a'' = ''r'', and ''p''<math>{} = 6R = 4r\sqrt{3}</math>, so
:<math>\begin{align}
  A &= \frac{ap}{2} \\
    &= \frac{r \cdot 4r\sqrt{3}}{2} = 2r^2\sqrt{3} \\
    &\approx 3.464 r^2.
\end{align}</math>

The regular hexagon fills the fraction <math>\tfrac{3\sqrt{3}}{2\pi} \approx 0.8270</math> of its [[circumscribed circle]].

If a regular hexagon has successive vertices A, B, C, D, E, F and if P is any point on the circumcircle between B and C, then {{nowrap|PE + PF {{=}} PA + PB + PC + PD}}.

It follows from the ratio of [[circumradius]] to [[inradius]] that the height-to-width ratio of a regular hexagon is 1:1.1547005; that is, a hexagon with a long [[diagonal]] of 1.0000000 will have a distance of 0.8660254 or cos(30°) between parallel sides.

=== Point in plane ===
For an arbitrary point in the plane of a regular hexagon with circumradius <math>R</math>, whose distances to the centroid of the regular hexagon and its six vertices are <math>L</math> and <math>d_i</math> 
respectively, we have<ref name=Mamuka >{{cite journal| last1= Meskhishvili |first1= Mamuka| date=2020|title=Cyclic Averages of Regular Polygons and Platonic Solids |journal= Communications in Mathematics and Applications|volume=11|pages=335–355|doi= 10.26713/cma.v11i3.1420|doi-broken-date= 1 November 2024|arxiv= 2010.12340|url= https://www.rgnpublications.com/journals/index.php/cma/article/view/1420/1065}}</ref>

:<math> d_1^2 + d_4^2 = d_2^2 + d_5^2 = d_3^2+ d_6^2= 2\left(R^2 + L^2\right), </math>
:<math> d_1^2 + d_3^2+ d_5^2 = d_2^2 + d_4^2+ d_6^2 = 3\left(R^2 + L^2\right), </math>
:<math> d_1^4 + d_3^4+ d_5^4 = d_2^4 + d_4^4+ d_6^4 = 3\left(\left(R^2 + L^2\right)^2 + 2 R^2 L^2\right). </math>

If <math>d_i</math> are the distances from the vertices of a regular hexagon to any point on its circumcircle, then <ref name= Mamuka />

:<math>\left(\sum_{i=1}^6 d_i^2\right)^2 = 4 \sum_{i=1}^6 d_i^4 .</math>

=== Construction ===
{{multiple image
| align = center
| image1 = Regular Hexagon Inscribed in a Circle.gif
| width1 = 240 
| alt1 = 
| caption1 = A step-by-step animation of the construction of a regular hexagon using [[compass and straightedge]], given by [[Euclid]]'s ''[[Euclid's Elements|Elements]]'', Book IV, Proposition 15: this is possible as 6 <math>=</math> 2 × 3, a product of a power of two and distinct [[Fermat prime]]s.
| image2 = 01-Sechseck-Seite-vorgegeben-wiki.svg
| width2 = 263
| alt2 = 
| caption2 = When the side length {{Overline|AB}} is given, drawing a circular arc from point A and point B gives the [[intersection]] M, the center of the [[circumscribed circle]]. Transfer the [[line segment]] {{Overline|AB}} four times on the circumscribed circle and connect the corner points.
| footer = 
}}

=== 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.

=== Tessellations ===
Like [[square (geometry)|square]]s and [[equilateral triangle]]s, regular hexagons fit together without any gaps to ''tile the plane'' (three hexagons meeting at every vertex), and so are useful for constructing [[tessellation]]s.<ref>{{cite book
 | first = Maciej | last = Dunajski
 | year = 2022
 | publisher = Oxford University Press
 | title = Geometry: A Very Short Introduction
 | url = https://books.google.com/books?id=zyRXEAAAQBAJ&pg=PA26
 | page = 26
 | isbn = 978-0-19-968368-0
}}</ref> The cells of a [[beehive (beekeeping)|beehive]] [[honeycomb]] are hexagonal for this reason and because the shape makes efficient use of space and building materials. The [[Voronoi diagram]] of a regular triangular lattice is the honeycomb tessellation of hexagons.