Hexagonal tiling

Hexagonal tiling
Type	Regular tiling
Vertex configuration	6.6.6 (or 63)
Schläfli symbol(s)	{6,3} t0,1{3,6}
Wythoff symbol(s)	3 | 6 2 2 6 | 3 3 3 3 |
Coxeter-Dynkin(s)
Symmetry	p6m, [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

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

There are 3 distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions.

k-uniform	1-uniform			2-uniform
Picture
Schläfli symbol	{6,3}	t{3,6}	t0,1,2{3[3]}
Wythoff symbol	3 | 6 2	2 6 | 3	3 3 3 |
symmetry	*632 (p6m)	*632 (p6m)	*333 (p3)	*632 (p6m)	*632 (p6m)
Coxeter-Dynkin diagram
Conway polyhedron notation	H	tH	teH	t6daH	t6dateH

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

Topologically identical tilings

Bricks as nonregular hexagonal tiling

The hexagonal tiling can be stretched and adjusted to other geometric proportions and different symmetries.

The standard brick pattern can be considered a nonregular hexagonal tiling. Each rectangular brick has vertices inserted on the two long edges, dividing them into two collinear edges.

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. The herringbone pattern is also a distorted hexagonal tiling.

4-color hexagonal tilings

Regular hexagons	Hexagonal weave
p6m (*632)	p6 (632)
Brick pattern	Herringbone
p4g (4*2)
|

Related polyhedra and tilings

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

(33)

(43)

(53)

(63) tiling

(73) tiling

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

(3.6.6)

(4.6.6)

(5.6.6)

(6.6.6) tiling

(7.6.6) tiling

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

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.)

Tiling	Schläfli symbol	Wythoff symbol	Vertex figure	Image
Hexagonal tiling	t0{6,3}	3 | 6 2	63
Truncated hexagonal tiling	t0,1{6,3}	2 3 | 6	3.12.12
Rectified hexagonal tiling (Trihexagonal tiling)	t1{6,3}	2 | 6 3	(3.6)2
Bitruncated hexagonal tiling (Truncated triangular tiling)	t1,2{6,3}	2 6 | 3	6.6.6
Dual hexagonal tiling (Triangular tiling)	t2{6,3}	6 | 3 2	36
Cantellated hexagonal tiling (Rhombitrihexagonal tiling)	t0,2{6,3}	6 3 | 2	3.4.6.4
Omnitruncated hexagonal tiling (Truncated trihexagonal tiling)	t0,1,2{6,3}	6 3 2 |	4.6.12
Snub hexagonal tiling	s{6,3}	| 6 3 2	3.3.3.3.6

See also

• Hexagonal lattice
• Hexagonal prismatic honeycomb
• Tilings of regular polygons
• List of uniform tilings
• List of regular polytopes

References

• 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 0-716-71193-1.  (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

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