Hexagonal tiling#Related tilings

{{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

}}

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {{math|{6,3} }} or {{math|t{3,6} }} (as a truncated triangular tiling).

English mathematician 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 three regular tilings of the plane. The other two are the triangular tiling and the square tiling.

Structure and properties

The hexagonal tiling has a structure consisting of a regular hexagon only as its prototile, sharing two vertices with other identical ones, an example of monohedral tiling.{{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]

}} Each vertex at the tiling is surrounded by three regular hexagons, denoted as $6.6.6$ by vertex configuration.{{cite book

| last1 = Grünbaum | first1 = Branko | author-link1 = Branko Grunbaum

| 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]

}} The dual of a hexagonal tiling is triangular tiling, because the center of each hexagonal tiling connects to another center of one, forming equilateral triangles.{{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

}}

Every mutually incident vertex, edge, and tile of a hexagonal tiling can be 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..{{sfnp|Grünbaum|Shephard|1987|p=[https://books.google.com/books?id=0x0vDAAAQBAJ&pg=PA35 35]}} The 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 is arranged like the hexagons in this tiling]]

If a circle is inscribed in each hexagon, the resulting figure is the densest way to arrange circles in two dimensions; its packing density is $\frac{\pi}{2\sqrt{3}} \approx 0.907$ .{{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}} The honeycomb theorem states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter.{{cite journal

| last = Hales | first = Thomas C. | author-link = Thomas Callister Hales

| 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

}} The optimal three-dimensional structure for making honeycomb (or rather, soap bubbles) was investigated by 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 = Graphene

| image2 =

| caption2 =

| image3 = Chicken Wire close-up.jpg

| caption3 = Chicken wire fencing

| total_width = 300

}}

The hexagonal tiling is commonly found in nature, such as the sheet of graphene with strong covalent carbon bonds. Tubular graphene sheets have been synthesised, known as carbon nanotubes.{{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 }} They have many potential applications, due to their high tensile strength and electrical properties. Silicene has a similar structure as graphene.

Chicken wire consists of a hexagonal lattice of wires, although the shape is not regular.{{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 = University of Chicago Press

| isbn = 978-0-226-33242-0

}}

File:

File:|

File:

File:Carbon nanotube zigzag povray.PNG|A 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 trylinka pavement crumbling in New York

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 copper, amongst other materials, forms a face-centered cubic lattice.

Uniform colorings

There are three distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions. 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}_h,k, and can be applied to hyperbolic tilings for p > 6.

class="wikitable" width=640 !k-uniform !colspan=3\|1-uniform !colspan=2\|2-uniform !colspan=2\|3-uniform
align=center !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 \|{6,3} \|t{3,6} \|t{3^[3]} \|colspan=2\| \|colspan=2\|
align=center !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 \|H \|tΔ \| \|cH=t6daH \| \|wH=t6dsH \|

class="wikitable" width=640

!k-uniform

!colspan=3|1-uniform

!colspan=2|2-uniform

!colspan=2|3-uniform

align=center

!Symmetry

|colspan=2|p6m, (*632)

| p3m1, (*333)

|colspan=2|p6m, (*632)

|colspan=2|p6, (632)

Picture

|100px

align=center

!Colors

align=center

!(h,k)

|(1,0)

|colspan=2|(1,1)

|colspan=2|(2,0)

|colspan=2|(2,1)

align=center

!Schläfli

|{6,3}

|t{3,6}

|t{3^[3]}

|colspan=2|

align=center

!Wythoff

| 3 {{pipe}} 6 2

| 2 6 {{pipe}} 3

| 3 3 3 {{pipe}}

|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|

align=center

!Conway

|tΔ

|cH=t6daH

|wH=t6dsH

The 3-color tiling is a tessellation generated by the order-3 permutohedrons.

= Chamfered hexagonal tiling=

A 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 at the limit]]

class=wikitable

!Hexagons (H)

!colspan=3|Chamfered hexagons (cH)

!Rhombi (daH)

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:

class=wikitable

!Regular tiling

!Dissection

!colspan=2|2-uniform tilings

!Regular tiling

!Inset

!Dual Tilings

align=center

|200x200px
Original

!100x100px 100x100px

|200x200px
1/3 dissected

|200x200px
2/3 dissected

|200x200px
fully dissected

!200x200px

|201x201px
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 and the rhombo-hexagonal dodecahedron tessellations in 3 dimensions.

class=wikitable

align=center

|160px
Rhombic tiling

|160px
Hexagonal tiling

|160px
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
Pentagonal tiling type 1 with overlays of regular hexagons (each comprising 2 pentagons).

|175px
pentagonal tiling type 3 with overlays of regular hexagons (each comprising 3 pentagons).

|175px
Pentagonal tiling type 4 with overlays of semiregular hexagons (each comprising 4 pentagons).

|175px
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 hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram {{CDD|node_1|6|node|n|node}}, progressing to infinity.

This tiling is topologically related to regular polyhedra with vertex figure n³, as a part of a sequence that continues into the hyperbolic plane.

It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6.

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.

Monohedral convex hexagonal tilings

There are 3 types of monohedral convex hexagonal tilings.Tilings and patterns, Sec. 9.3 Other Monohedral tilings by convex polygons They are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains glide reflections, and is 2-isohedral keeping chiral pairs distinct.

p2, 2222\|\|pgg, 22×\|\|p2, 2222\|\|p3, 333
class=wikitable \|+ 3 types of monohedral convex hexagonal tilings !1	colspan=2\|2	3
120px	\|120px	120px	\|120px
align=center \|120px b = e B + C + D = 360° \|colspan=2\|120px b = e, d = f B + C + E = 360° \|120px a = f, b = c, d = e B = D = F = 120°
align=center \|120px 2-tile lattice \|colspan=2\|120px 4-tile lattice \|120px 3-tile lattice

There are also 15 monohedral convex pentagonal tilings, 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.Tilings and patterns, from list of 107 isohedral tilings, pp. 473–481 Single-color (1-tile) lattices are 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
Parallelogram

|100px
Trapezoid

|100px
Parallelogram

|70px
Rectangle

|100px
Parallelogram

|100px
Rectangle

class=wikitable

|+ Isohedrally-tiled pentagons

align=center

!p2 (2222)

pgg (22×)

p3 (333)

align=center

|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.Tilings and patterns, uniform tilings that are not edge-to-edge

It can also be distorted into a chiral 4-colored tri-directional weaved pattern, distorting some hexagons into parallelograms. The weaved pattern with 2 colored faces has rotational 632 (p6) symmetry. A 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

class=wikitable

Regular

!Gyrated

!Regular

!Weaved

!Chevron

p6m, (*632)

! p6, (632)

! p6m (*632)

! p6 (632)

! p1 (°)

150px

|150px

p3m1, (*333)

! p3, (333)

! p6m (*632)

! p2 (2222)

! p1 (°)

150px

|150px

Circle packing

The hexagonal tiling can be used as a 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).Order in Space: A design source book, Keith Critchlow, pp. 74–75, pattern 2 The gap inside each hexagon allows for one circle, creating the densest packing from the 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.Coxeter, Regular Complex Polytopes, pp. 111–112, p. 136.

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}}

class=wikitable

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}}

References

Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, {{ISBN|0-486-61480-8}} p. 296, Table II: Regular honeycombs
{{cite book | author1=Grünbaum, Branko | author-link=Branko Grünbaum | 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. 58–65)
{{The Geometrical Foundation of Natural Structure (book)|page=35}}
John H. Conway, Heidi Burgiel, 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]