View source for Hexagonal tiling

{{short description|Regular tiling of a two-dimensional space}}
{{More footnotes|date=March 2011}}
{{Infobox face-uniform tiling
 | name = Hexagonal tiling
 | image = Tiling Regular 6-3 Hexagonal.svg
 | type = regular tiling
 | tile = regular hexagon
 | vertex_config = 6.6.6
 | symmetry = p6m
 | properties = vertex-transitive, edge-transitive, face-transitive
 | dual = [[Triangular tiling|triangular tiling]]
}}

In [[Geometry|geometry]], the '''hexagonal tiling''' or '''hexagonal tessellation''' is a regular tiling of the [[Euclidean plane]], in which exactly three [[Hexagon|hexagon]]s meet at each vertex. It has [[Schläfli symbol]] of {{math|{6,3} }} or {{math|''t''{3,6} }} (as a [[Truncation (geometry)|truncated]] triangular tiling).

English mathematician [[Biography:John Horton Conway|John Conway]] called it a '''hextille'''.

The internal angle of the hexagon is 120 degrees, so three hexagons at a point make a full 360 degrees. It is one of [[List of regular polytopes#Euclidean tilings|three regular tilings of the plane]]. The other two are the [[Triangular tiling|triangular tiling]] and the [[Square tiling|square tiling]].

== Structure and properties ==
The hexagonal tiling has a structure consisting of a regular hexagon only as its [[Prototile|prototile]], sharing two vertices with other identical ones, an example of monohedral tiling.<ref name=adams>{{cite book
 | title = The Tiling Book: An Introduction to the Mathematical Theory of Tilings
 | first = Colin | last = Adams
 | publisher = American Mathematical Society
 | year = 2022
 | isbn = 9781470468972
 | pages = [https://books.google.com/books?id=LvGGEAAAQBAJ&pg=PA23 23]
}}</ref> Each vertex at the tiling is surrounded by three regular hexagons, denoted as <math> 6.6.6 </math> by [[Vertex configuration|vertex configuration]].<ref name=gs>{{cite book
 | last2 = Shephard | first2 = G. C.
 | year = 1987
 | title = Tilings and Patterns
 | title-link = Tilings and patterns
 | publisher = W. H. Freeman
 | page = [https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA21 21]
}}</ref> The dual of a hexagonal tiling is [[Triangular tiling|triangular tiling]], because the center of each hexagonal tiling connects to another center of one, forming [[Equilateral triangle|equilateral triangle]]s.<ref name=ns>{{cite journal
 | last1 = Nelson | first1 = Roice
 | last2 = Segerman | first2 = Henry
 | title = Visualizing hyperbolic honeycombs
 | journal = Journal of Mathematics and the Arts
 | year = 2017
 | volume = 11 | issue = 1 | pages = 4–39
 | doi = 10.1080/17513472.2016.1263789
 | arxiv = 1511.02851
}}</ref>

Every mutually incident vertex, edge, and tile of a hexagonal tiling can be [[Group action (mathematics)#Types of actions|act transitively]] to another of those three by mapping the first ones to the second through the symmetry operation. In other words, they are vertex-transitive (mapping the vertex of a tile to another), edge-transitive (mapping the edge to another), and face-transitive (mapping regular hexagonal tile to another). From these, the hexagonal tiling is categorized as one of three regular tilings; the remaining being its dual and [[Square tiling|square tiling]].{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA35 35]}} The [[Symmetry group|symmetry group]] of a hexagonal tiling is p6m.{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA42 42]|loc=see p. [https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA38 38] for detail of symbols}}

== Applications ==
[[File:Kissing-2d.svg|thumb|upright=0.6|The densest [[Circle packing|circle packing]] is arranged like the hexagons in this tiling]]
If a circle is inscribed in each hexagon, the resulting figure is the densest way to [[Sphere packing|arrange circles]] in two dimensions; its [[Packing density|packing density]] is <math display="inline"> \frac{\pi}{2\sqrt{3}} \approx 0.907 </math>.<ref>{{cite arXiv|last1=Chang|first1=Hai-Chau|last2=Wang|first2=Lih-Chung|title=A Simple Proof of Thue's Theorem on Circle Packing|eprint=1009.4322|date=22 September 2010|class=math.MG}}</ref> The [[Honeycomb theorem|honeycomb theorem]] states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter.<ref>{{cite journal
 | date = January 2001
 | title = The Honeycomb Conjecture
 | journal = Discrete and Computational Geometry
 | volume = 25 | issue = 1 | pages = 1–22
 | arxiv = math/9906042
 | doi = 10.1007/s004540010071
 | mr = 1797293
 | s2cid = 14849112
}}</ref> The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by [[Biography:Lord Kelvin|Lord Kelvin]], who believed that the Kelvin structure (or body-centered cubic lattice) is optimal. However, the less regular [[Weaire–Phelan structure]] is slightly better.

{{multiple image
 | image1 = Graphene xyz.jpg
 | caption1 = [[Physics:Graphene|Graphene]]
 | image2 = 
 | caption2 =
 | image3 = Chicken Wire close-up.jpg
 | caption3 = [[Chemistry:Chicken wire|Chicken wire]] fencing
 | total_width = 300
}}
The hexagonal tiling is commonly found in nature, such as the sheet of [[Physics:Graphene|graphene]] with strong covalent carbon bonds. Tubular graphene sheets have been synthesised, known as [[Physics:Carbon nanotube|carbon nanotube]]s.<ref name=christensen>{{cite book
 | title = The Theory of Materials Failure
 | first = Richard M. | last = Christensen
 | year = 2013
 | publisher = Oxford University Press
 | url = https://books.google.com/books?id=MwxGWnG7t58C&pg=PA201
 | page = 201
| isbn = 978-0-19-966211-1 }}</ref> They have many potential applications, due to their high [[Physics:Tensile strength|tensile strength]] and electrical properties. [[Chemistry:Silicene|Silicene]] has a similar structure as graphene.

[[Chemistry:Chicken wire|Chicken wire]] consists of a hexagonal lattice of wires, although the shape is not regular.<ref name=ball>{{cite book
 | last = Ball | first = Philip
 | year = 2016
 | title = Patterns in Nature: Why the Natural World Looks the Way It Does
 | url = https://books.google.com/books?id=hm70CwAAQBAJ&pg=PA15
 | page = 15
 | publisher = [[Organization:University of Chicago Press|University of Chicago Press]]
| isbn = 978-0-226-33242-0
 }}</ref>

<gallery>
File:
File:|
File:
File:Carbon nanotube zigzag povray.PNG|A [[Physics:Carbon nanotube|carbon nanotube]] can be seen as a hexagon tiling on a cylindrical surface
File:Tile (AM 1955.117-1).jpg|alt=Hexagonal tile with blue bird and flowers|Hexagonal Persian tile {{circa|1955}}
File:Hex pavers sliding to Hudson W60 jeh.jpg|Hexagonal&nbsp;[[Engineering:Trylinka|trylinka]] pavement crumbling in New York
</gallery>

The hexagonal tiling appears in many crystals. In three dimensions, the face-centered cubic and hexagonal close packing are common crystal structures. They are the densest sphere packings in three dimensions. Structurally, they comprise parallel layers of hexagonal tilings, similar to the structure of graphite. They differ in the way that the layers are staggered from each other, with the face-centered cubic being the more regular of the two. Pure [[Chemistry:Copper|copper]], amongst other materials, forms a face-centered cubic lattice.

{{Clear}}

== Uniform colorings ==
There are three distinct [[Uniform coloring|uniform coloring]]s of a hexagonal tiling, all generated from reflective symmetry of [[Wythoff construction]]s. The (''h'',''k'') represent the periodic repeat of one colored tile, counting hexagonal distances as ''h'' first, and ''k'' second. The same counting is used in the Goldberg polyhedra, with a notation {''p''+,3}<sub>''h'',''k''</sub>, and can be applied to hyperbolic tilings for ''p''&nbsp;>&nbsp;6.
{| class="wikitable" width=640
!''k''-uniform
!colspan=3|1-uniform
!colspan=2|2-uniform
!colspan=2|3-uniform
|- align=center
![[List of planar symmetry groups|Symmetry]]
|colspan=2|p6m, (*632)
| p3m1, (*333)
|colspan=2|p6m, (*632)
|colspan=2|p6, (632)
|-
!Picture
|100px
|100px
|100px
|100px
|100px
|100px
|100px
|- align=center
!Colors
!1
!2
!3
!2
!4
!2
!7
|- align=center
!(h,k)
|(1,0)
|colspan=2|(1,1)
|colspan=2|(2,0)
|colspan=2|(2,1)
|- align=center
![[Schläfli symbol|Schläfli]]
|{6,3}
|t{3,6}
|t{3<sup>[3]</sup>}
|colspan=2|
|colspan=2|
|- align=center
![[Wythoff symbol|Wythoff]]
| 3 {{pipe}} 6 2
| 2 6 {{pipe}} 3
| 3 3 3 {{pipe}}
|colspan=2|
|colspan=2|
|- align=center
!Coxeter
|{{CDD|node_1|6|node|3|node}}
|{{CDD|node_1|3|node_1|6|node}}
|{{CDD|node_1|split1|branch_11}}
|colspan=2|
|colspan=2|
|- align=center
![[Conway polyhedron notation|Conway]]
|H
|tΔ
|
|cH=t6daH
|
|wH=t6dsH
|
|}

The 3-color tiling is a tessellation generated by the order-3 [[Permutohedron|permutohedron]]s.

=== Chamfered hexagonal tiling===

A [[Chamfer (geometry)|chamfered]] hexagonal tiling replaces edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling.
[[File:ChamferedHexTilingAnimation.gif|thumb|The ''chamfered hexagonal tiling'' degenerates into a [[rhombille tiling]] at the limit]]
{| class=wikitable
!Hexagons (H)
!colspan=3|Chamfered hexagons (cH)
!Rhombi (daH)
|-
|120px
|120px
|120px
|120px
|120px
|}

== Related tilings ==

The hexagons can be dissected into sets of 6 triangles. This process leads to two 2-uniform tilings, and the [[Triangular tiling|triangular tiling]]:
{| class=wikitable
!Regular tiling
!Dissection
!colspan=2|2-uniform tilings
!Regular tiling
!Inset
!Dual Tilings
|- align=center
|200x200px<br/>Original
!100x100px100x100px
|200x200px<br/>1/3 dissected
|200x200px<br/>2/3 dissected
|200x200px<br/>fully dissected
!200x200px
|201x201px<br />E to IH to FH to H
|}

The hexagonal tiling can be considered an ''elongated rhombic tiling'', where each vertex of the rhombic tiling is stretched into a new edge. This is similar to the relation of the [[Rhombic dodecahedron#Honeycomb|rhombic dodecahedron]] and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions.
{| class=wikitable
|- align=center
|160px<br/>Rhombic tiling
|160px<br/>Hexagonal tiling
|160px<br/>Fencing uses this relation
|}

It is also possible to subdivide the prototiles of certain hexagonal tilings by two, three, four or nine equal pentagons:
{| class=wikitable width=700
|- valign=top
|175px<br/>[[Pentagonal tiling]] type 1 with overlays of regular hexagons (each comprising 2 pentagons).
|175px<br/>pentagonal tiling type 3 with overlays of regular hexagons (each comprising 3 pentagons).
|175px<br/>Pentagonal tiling type 4 with overlays of semiregular hexagons (each comprising 4 pentagons).
|175px<br/>Pentagonal tiling type 3 with overlays of two sizes of regular hexagons (comprising 3 and 9 pentagons respectively).
|}

=== Symmetry mutations===

This tiling is topologically related as a part of a sequence of regular tilings with [[Hexagon|hexagon]]al faces, starting with the hexagonal tiling, with [[Schläfli symbol]] {6,n}, and [[Coxeter diagram]] {{CDD|node_1|6|node|n|node}}, progressing to infinity.
{{Hexagonal_regular_tilings}}

This tiling is topologically related to regular polyhedra with [[Vertex figure|vertex figure]] ''n''<sup>3</sup>, as a part of a sequence that continues into the [[Hyperbolic space|hyperbolic plane]].
{{Order-3 tiling table}}

It is similarly related to the uniform [[Truncation (geometry)|truncated]] polyhedra with vertex figure ''n''.6.6.
{{Truncated figure2 table}}

This tiling is also part of a sequence of truncated rhombic polyhedra and tilings with [n,3] [[Coxeter group]] symmetry. The cube can be seen as a rhombic hexahedron where the rhombi are squares. The truncated forms have regular n-gons at the truncated vertices, and nonregular hexagonal faces.
{{Dual quasiregular3 table}}

== Monohedral convex hexagonal tilings ==
There are 3 types of monohedral convex hexagonal tilings.<ref>''[[Tilings and patterns]]'', Sec. 9.3 Other Monohedral tilings by convex polygons</ref> They are all [[Isohedral figure|isohedral]]. Each has parametric variations within a fixed symmetry. Type 2 contains [[Glide reflection|glide reflection]]s, and is 2-isohedral keeping chiral pairs distinct.
{| class=wikitable
|+ 3 types of monohedral convex hexagonal tilings
!1||colspan=2|2||3
|-
!p2, 2222||pgg, 22×||p2, 2222||p3, 333
|-
|120px|||120px||120px|||120px
|- align=center
|120px<BR/>b = e<BR/>B + C + D = 360°
|colspan=2|120px<BR/>b = e, d = f<BR/>B + C + E = 360°
|120px<BR/>a = f, b = c, d = e<BR/>B = D = F = 120°
|- align=center
|120px<BR/>2-tile lattice
|colspan=2|120px<BR/>4-tile lattice
|120px<BR/>3-tile lattice
|}

There are also 15 monohedral convex [[Pentagonal tiling|pentagonal tiling]]s, as well as all quadrilaterals and triangles.

== Topologically equivalent tilings ==

Hexagonal tilings can be made with the identical {6,3} topology as the regular tiling (3 hexagons around every vertex). With isohedral faces, there are 13 variations. Symmetry given assumes all faces are the same color. Colors here represent the lattice positions.<ref>''[[Tilings and patterns]]'', from list of 107 isohedral tilings, pp.&nbsp;473–481</ref> Single-color (1-tile) lattices are [[Parallelogon|parallelogon]] hexagons. 
{| class=wikitable
|+ 13 isohedrally-tiled hexagons
|- align=center
!colspan=2|pg (××)||p2 (2222)||p3 (333)||colspan=2|pmg (22*)
|- align=center
|100px
|100px
|100px
|100px
|100px
|100px
|- align=center
!colspan=3|pgg (22×)||p31m (3*3)||p2 (2222)||cmm (2*22)||p6m (*632)
|- align=center
|100px
|100px
|100px
|100px
|100px
|100px
|100px
|}

Other isohedrally-tiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edge-to-edge, but interpreted as colinear adjacent edges:
{| class=wikitable
|+ Isohedrally-tiled quadrilaterals
|- align=center
!colspan=2|pmg (22*)||colspan=3|pgg (22×)||cmm (2*22)||p2 (2222)
|- align=center
|100px<br/>Parallelogram
|100px<br/>[[Trapezoid]]
|100px<br/>Parallelogram
|70px<br/>[[Rectangle]]
|100px<br/>[[Parallelogram]]
|100px<br/>Rectangle
|100px<br/>Rectangle
|}
{| class=wikitable
|+ Isohedrally-tiled pentagons
|- align=center
!p2 (2222)||pgg (22×)||p3 (333)
|- align=center
|120px
||120px
||120px
|}

The 2-uniform and 3-uniform tessellations have a rotational degree of freedom which distorts 2/3 of the hexagons, including a colinear case that can also be seen as a non-edge-to-edge tiling of hexagons and larger triangles.<ref>''[[Tilings and patterns]]'', uniform tilings that are not edge-to-edge</ref>

It can also be distorted into a [[Chirality (mathematics)|chiral]] 4-colored tri-directional weaved pattern, distorting some hexagons into [[Parallelogram|parallelogram]]s. The weaved pattern with 2 colored faces has rotational [[List of planar symmetry groups#Wallpaper groups|632 (p6) symmetry]]. A [[Engineering:Chevron (insignia)|chevron]] pattern has pmg (22*) symmetry, which is lowered to p1 (°) with 3 or 4 colored tiles.

{| class=wikitable
|-
!Regular
!Gyrated
!Regular
!Weaved
!Chevron
|-
! p6m, (*632)
! p6, (632)
! p6m (*632)
! p6 (632)
! p1 (°)
|-
|150px
|150px
|150px
|150px
|150px
|-
! p3m1, (*333)
! p3, (333)
! p6m (*632)
! p2 (2222)
! p1 (°)
|-
|150px
|150px
|150px
|150px
|150px
|}

== Circle packing ==
[[File:comparison_hex_tiling.svg|thumb|upright=0.5|Hexagonal (mid&shy;dle) vs equivalent circle (top) vs staggered rectan&shy;gular tilings (bottom)]]
The hexagonal tiling  can be used as a [[Circle packing|circle packing]], placing equal-diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing ([[Kissing number|kissing number]]).<ref name=Critchlow>Order in Space: A design source book, Keith Critchlow, pp.&nbsp;74–75, pattern 2</ref> <!--The packing density is ... % coverage.)--> The gap inside each hexagon allows for one circle, creating the densest packing from the [[Triangular tiling#A2 lattice and circle packings|triangular tiling]], with each circle in contact with a maximum of 6 circles.
:200px

== Related regular complex apeirogons ==

There are 2 regular complex apeirogons, sharing the vertices of the hexagonal tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons ''p''{''q''}''r'' are constrained by: 1/''p'' + 2/''q'' + 1/''r'' = 1. Edges have ''p'' vertices, and vertex figures are ''r''-gonal.<ref>Coxeter, Regular Complex Polytopes, pp. 111–112, p. 136.</ref>

The first is made of 2-edges, three around every vertex, the second has hexagonal edges, three around every vertex. A third complex apeirogon, sharing the same vertices, is quasiregular, which alternates 2-edges and 6-edges.
{| class=wikitable
|-
|160px
|160px
|160px
|-
!2{12}3 or {{CDD|node_1|12|3node}}
!6{4}3 or {{CDD|6node_1|4|3node}}
!{{CDD|6node_1|6|node_1}}
|}

==See also==
*[[Hexagonal lattice]]
*Hexagonal prismatic honeycomb
*Tilings of regular polygons
*List of uniform tilings
*[[List of regular polytopes]]
* [[Hexagonal tiling honeycomb]]
*[[Hex map]] board game design

== References ==
{{Reflist}}
* Coxeter, H.S.M. ''Regular Polytopes'', (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} p.&nbsp;296, Table II: Regular honeycombs
* {{cite book | author1=Grünbaum, Branko |  author2=Shephard, G. C. | title=Tilings and Patterns | location=New York | publisher=W. H. Freeman | year=1987 | isbn=0-7167-1193-1 | url-access=registration | url=https://archive.org/details/isbn_0716711931 }} (Chapter 2.1: ''Regular and uniform tilings'', pp.&nbsp;58–65)
* {{The Geometrical Foundation of Natural Structure (book)|page=35}}
* John H. Conway, Heidi Burgiel, [[Biography:Chaim Goodman-Strauss|Chaim Goodman-Strauss]], ''The Symmetries of Things'' 2008, {{ISBN|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205]

== External links ==
* {{MathWorld | urlname=HexagonalGrid | title=Hexagonal Grid}}
** {{MathWorld | urlname=RegularTessellation | title=Regular tessellation}}
** {{MathWorld | urlname=UniformTessellation | title=Uniform tessellation}}
* {{KlitzingPolytopes|flat.htm#2D|2D Euclidean tilings|o3o6x – hexat – O3}}

{{Honeycombs}}
{{Tessellation}}

[[Category:Euclidean tilings]]
[[Category:Hexagonal tilings| ]]
[[Category:Isogonal tilings]]
[[Category:Isohedral tilings]]
[[Category:Regular tilings]]
[[Category:Regular tessellations]]

{{Sourceattribution|Hexagonal tiling}}