# Hexagonal tiling

Hexagonal tiling

Type Regular tiling
Vertex configuration 6.6.6 (or 63)
Schläfli symbol(s) {6,3}
t{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

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 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. 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{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

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

Spherical
Polyhedra
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 Compact hyperbolic Paracompact
*232
[2,3]
D3h
*332
[3,3]
Td
*432
[4,3]
Oh
*532
[5,3]
Ih
*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{3,2}

t{3,3}

t{3,4}

t{3,5}

t{3,6}

t{3,7}

t{3,8}

t{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

Chamfered tetrahedron

Chamfered cube

Chamfered dodecahedron

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.)

Uniform hexagonal/triangular tilings
Symmetry: [6,3], (*632) [6,3]+
(632)
[1+,6,3]
(*333)
[6,3+
(3*3)
{6,3} t{6,3} r{6,3}
r{3[3]}
t{3,6}
t{3[3]}
{3,6}
{3[3]}
rr{6,3}
s2{6,3}
tr{6,3} sr{6,3} h{6,3}
{3[3]}
h2{6,3}
r{3[3]}
s{3,6}
s{3[3]}

=

=

=
=
or
=
or

=

Uniform duals
V63 V3.122 V(3.6)2 V63 V36 V3.4.12.4 V.4.6.12 V34.6 V36 V(3.6)2 V36

Triangle
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
a1 [3[3] ×1 (None)
i2 <[3[3]>
= [6,3]

=
×2 1, 2
r6 [3[3[3]]
= [6,3]

=
×6 3, (1)
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

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

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

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.