Hexagonal tiling

Hexagonal tiling

Type	Regular tiling
Vertex configuration	6.6.6 (or 6³)
Schläfli symbol(s)	{6,3} t_0,1{3,6}
Wythoff symbol(s)	3 \| 6 2 2 6 \| 3 3 3 3 \|
Coxeter diagram(s)
Symmetry	p6m, [6,3], (*632)
Rotation symmetry	p6, [6,3]⁺, (632)
Dual	Triangular tiling
Properties	Vertex-transitive, edge-transitive, face-transitive
6.6.6 (or 6³)

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

Conway calls 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.

Applications[edit]

The hexagonal tiling is the densest way to arrange circles in two dimensions. The Honeycomb conjecture states that the hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter. The optimal three-dimensional structure for making beehives (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.

Chicken wire consists of a hexagonal lattice of wires. This structure exists naturally in the form of graphite, where each sheet of graphene resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised; these are known as carbon nanotubes. They have many potential applications, due to their high tensile strength and electrical properties.

The densest circle packing is arranged like the hexagons in this tiling
Chicken wire fencing
Graphene
carbon nanotube can be seen as a hexagon tiling on a cylindrical surface

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 known sphere packings in three dimensions, and are believed to be optimal. 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[edit]

There are 3 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.

k-uniform	1-uniform			2-uniform		3-uniform
Picture
Colors	1	2	3	2	4	2	7
(h,k)	(1,0)	(1,1)		(2,0)		(2,1)
Schläfli symbol	{6,3}	t{3,6}	t_0,1,2{3^[3]}
Wythoff symbol	3 \| 6 2	2 6 \| 3	3 3 3 \|
Symmetry	*632 (p6m) [6,3]		*333 (p3) [3^[3]]	*632 (p6m) [6,3]		632 (p6) [6,3]⁺
Coxeter-Dynkin diagram
Conway polyhedron notation	H	tH	teH	t6daH	t6dateH

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

Related polyhedra and tilings[edit]

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

Polyhedra				Euclidean	Hyperbolic tilings
{2,3}	{3,3}	{4,3}	{5,3}	{6,3}	{7,3}	{8,3}	...	(∞,3}

This tiling is topologically related as a part of sequence of regular tilings with hexagonal faces, starting with the hexagonal tiling, with Schläfli symbol {6,n}, and Coxeter diagram , progressing to infinity.

Spherical	Euclidean	Hyperbolic tilings
{6,2}	{6,3}	{6,4}	{6,5}	{6,6}	{6,7}	{6,8}	...	{6,∞}

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

Dimensional family of truncated polyhedra and tilings: *n.6.6*
Symmetry *n42 [n,3]	Spherical				Euclidean	Hyperbolic...
Symmetry *n42 [n,3]	*232 [2,3] D_3h	*332 [3,3] T_d	*432 [4,3] O_h	*532 [5,3] I_h	*632 [6,3] P6m	*732 [7,3]	*832 [8,3]...	*∞32 [∞,3]
Order	12	24	48	120	∞
Truncated figures	2.6.6	3.6.6	4.6.6	5.6.6	6.6.6	7.6.6	8.6.6	∞.6.6
Coxeter Schläfli	t_0,1{3,2}	t_0,1{3,3}	t_0,1{3,4}	t_0,1{3,5}	t_0,1{3,6}	t_0,1{3,7}	t_0,1{3,8}	t_0,1{3,∞}
Uniform dual figures
n-kis figures	V2.6.6	V3.6.6	V4.6.6	V5.6.6	V6.6.6	V7.6.6	V8.6.6	V∞.6.6
Coxeter

This tiling is also a 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. The sequence has two vertex figures (n.6.6) and (6,6,6).

Polyhedra			Euclidean tiling	Hyperbolic tiling
[3,3]	[4,3]	[5,3]	[6,3]	[7,3]	[8,3]
Cube	Rhombic dodecahedron	Rhombic triacontahedron	Rhombille
Alternate truncated cube	Truncated rhombic dodecahedron	Truncated rhombic triacontahedron	Hexagonal tiling

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.

Rhombic tiling	Hexagonal tiling	Fencing uses this relation

Wythoff constructions from hexagonal and triangular tilings[edit]

Like the uniform polyhedra there are eight uniform tilings that can be based from the regular hexagonal tiling (or the dual triangular tiling).

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 which are topologically distinct. (The truncated triangular tiling is topologically identical to the hexagonal tiling.)

Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632)							[6,3]⁺, (632)	[1⁺,6,3], (*333)	[6,3⁺], (3*3)


{6,3}	t_0,1{6,3}	t₁{6,3}	t_1,2{6,3}	t₂{6,3}	t_0,2{6,3}	t_0,1,2{6,3}	s{6,3}	h{6,3}	h_1,2{6,3}
Uniform duals


V6.6.6	V3.12.12	V3.6.3.6	V6.6.6	V3.3.3.3.3.3	V3.4.12.4	V.4.6.12	V3.3.3.3.6	V3.3.3.3.3.3

Wythoff	3 \| 3 3	3 3 \| 3	3 \| 3 3	3 3 \| 3	3 \| 3 3	3 3 \| 3	3 3 3 \|	\| 3 3 3
Coxeter
Image Vertex figure	(3.3)³	3.6.3.6	(3.3)³	3.6.3.6	(3.3)³	3.6.3.6	6.6.6	3.3.3.3.3.3

Topologically identical tilings[edit]

Hexagonal tilings can be made with the identical {6,3} topology as the regular tiling (3 hexagons around every vertex). With identical faces (face-transitivity) and vertex-transitivity, there are 12 variations, with the first 7 identified as quadrilaterals that don't connect edge-to-edge, or as hexagons with two pairs of colinear edges. Symmetry given assumes all faces are the same color.^[1]

Parallelogram
p2 symmetry
Parallelogram
pmg symmetry
Parallelogram
pgg symmetry
rectangle
pgg symmetry
Trapezoid
pmg symmetry
Rectangle
pgg symmetry
Rectangle
cmm symmetry
Hexagon
p2 symmetry
Hexagon
pgg symmetry
Hexagon
pmg symmetry
Stretched hexagon
cmm symmetry
Regular hexagon
p6m symmetry

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

4-color hexagonal tilings
Regular hexagons	Hexagonal weave
p6m (*632)	p6 (632)

Circle packing[edit]

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). The gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling#circle packing, with each circle contact with the maximum of 6 circles.

References[edit]

^ Tilings and Patterns, from list of 107 isohedral tilings, p.473-481

Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p. 58-65)
Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 35. ISBN 0-486-23729-X.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1]

External links[edit]

Weisstein, Eric W., "Hexagonal Grid", MathWorld.
- Weisstein, Eric W., "Regular tessellation", MathWorld.
- Weisstein, Eric W., "Uniform tessellation", MathWorld.
Richard Klitzing, 2D Euclidean tilings, o3o6x - hexat - O3

[1] Tilings and Patterns, from list of 107 isohedral tilings, p.473-481

[1]

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–11
Family	${\tilde{A}}_{n-1}$	${\tilde{C}}_{n-1}$	${\tilde{B}}_{n-1}$ / ${\tilde{D}}_{n-1}$	${\tilde{E}}_{n-1}$ / ${\tilde{F}}_4$ / ${\tilde{G}}_2$
Uniform tiling	Triangular	Square		Hexagonal
Uniform convex honeycomb	Tetrahedral-octahedral	Cubic honeycomb	Tetrahedral-octahedral
Uniform 5-honeycomb	5-cell honeycomb	Tesseractic honeycomb	16-cell honeycomb	24-cell honeycomb
Uniform 6-honeycomb	5-simplex honeycomb	5-cube honeycomb	5-demicube honeycomb
Uniform 7-honeycomb	6-simplex honeycomb	6-cube honeycomb	6-demicube honeycomb	2₂₂ honeycomb
Uniform 8-honeycomb	7-simplex honeycomb	7-cubic honeycomb	7-demicube honeycomb	1₃₃ • 3₃₁ honeycombs
Uniform 9-honeycomb	8-simplex honeycomb	8-cubic honeycomb	8-demicube honeycomb	1₅₂ • 2₅₁ • 5₂₁ honeycombs
Uniform 10-honeycomb	9-simplex honeycomb	9-cube honeycomb	9-demicube honeycomb
Uniform 11-honeycomb	10-simplex honeycomb	10-cube honeycomb	10-demicube honeycomb
Uniform n-honeycomb	n-simplectic honeycomb	n-cubic honeycomb	n-demicubic honeycomb	1_k2 • 2_k1 • k₂₁ figures

Hexagonal tiling

Contents

Applications[edit]

Uniform colorings[edit]

Related polyhedra and tilings[edit]

Wythoff constructions from hexagonal and triangular tilings[edit]

Topologically identical tilings[edit]

Circle packing[edit]

See also[edit]

References[edit]

External links[edit]

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