Hexagonal tiling

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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) CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel split1.pngCDel branch 11.png
Symmetry p6m, [6,3], (*632)
Rotation symmetry p6, [6,3]+, (632)
Dual Triangular tiling
Properties Vertex-transitive, edge-transitive, face-transitive
Hexagonal tiling
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 Uniform tiling 63-t0.png Uniform tiling 63-t12.png Uniform tiling 333-t012.png Truncated rhombille tiling.png Hexagonal tiling 4-colors.svg Hexagonal tiling 2-1.png Hexagonal tiling 7-colors.png
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 CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node 1.pngCDel split1.pngCDel branch 11.png
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
Spherical trigonal hosohedron.png
{2,3}
CDel node 1.pngCDel 2.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-33-t0.png
{3,3}
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-43-t0.png
{4,3}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-53-t0.png
{5,3}
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png
Uniform polyhedron-63-t0.png
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 237-1.png
{7,3}
CDel node 1.pngCDel 7.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 238-1.png
{8,3}
CDel node 1.pngCDel 8.pngCDel node.pngCDel 3.pngCDel node.png
... H2 tiling 23i-1.png
(∞,3}
CDel node 1.pngCDel infin.pngCDel node.pngCDel 3.pngCDel node.png

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 CDel node 1.pngCDel 6.pngCDel node.pngCDel n.pngCDel node.png, progressing to infinity.

Spherical Euclidean Hyperbolic tilings
Hexagonal dihedron.png
{6,2}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 2.pngCDel node.png
Uniform tiling 63-t0.png
{6,3}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
H2 tiling 246-1.png
{6,4}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 4.pngCDel node.png
H2 tiling 256-1.png
{6,5}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 5.pngCDel node.png
H2 tiling 266-4.png
{6,6}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 6.pngCDel node.png
H2 tiling 267-4.png
{6,7}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 7.pngCDel node.png
H2 tiling 268-4.png
{6,8}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 8.pngCDel node.png
... H2 tiling 26i-4.png
{6,∞}
CDel node 1.pngCDel 6.pngCDel node.pngCDel infin.pngCDel node.png

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
Hexagonal dihedron.png
2.6.6
Uniform tiling 332-t12.png
3.6.6
Uniform tiling 432-t12.png
4.6.6
Uniform tiling 532-t12.png
5.6.6
Uniform tiling 63-t12.png
6.6.6
Uniform tiling 73-t12.png
7.6.6
Uniform tiling 83-t12.png
8.6.6
H2 tiling 23i-6.png
∞.6.6
Coxeter
Schläfli
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node.png
t{3,2}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png
t{3,3}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 4.pngCDel node.png
t{3,4}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 5.pngCDel node.png
t{3,5}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png
t{3,6}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 7.pngCDel node.png
t{3,7}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 8.pngCDel node.png
t{3,8}
CDel node 1.pngCDel 3.pngCDel node 1.pngCDel infin.pngCDel node.png
t{3,∞}
Uniform dual figures
n-kis
figures
Hexagonal hosohedron.png
V2.6.6
Triakistetrahedron.jpg
V3.6.6
Tetrakishexahedron.jpg
V4.6.6
Pentakisdodecahedron.jpg
V5.6.6
Uniform tiling 63-t2.png
V6.6.6
Order3 heptakis heptagonal til.png
V7.6.6
Uniform dual tiling 433-t012.png
V8.6.6
H2checkers 33i.png
V∞.6.6
Coxeter CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 2.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 5.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 6.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 7.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 8.pngCDel node.png CDel node f1.pngCDel 3.pngCDel node f1.pngCDel infin.pngCDel node.png

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]
Hexahedron.svg
Cube
Rhombicdodecahedron.jpg
Rhombic dodecahedron
Rhombictriacontahedron.jpg
Rhombic triacontahedron
Rhombic star tiling.png
Rhombille
Order73 qreg rhombic til.png Uniform dual tiling 433-t01-yellow.png
Alternate truncated cube.png
Chamfered tetrahedron
Truncated rhombic dodecahedron2.png
Chamfered cube
Truncated rhombic triacontahedron.png
Chamfered dodecahedron
Truncated rhombille tiling.png
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.

Kah 3 6 romb.png
Rhombic tiling
Uniform tiling 63-t0.png
Hexagonal tiling
Chicken Wire close-up.jpg
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]}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel branch 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel branch 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel branch.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png =
CDel branch 10ru.pngCDel split2.pngCDel node.png or CDel branch 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel branch 10ru.pngCDel split2.pngCDel node 1.png or CDel branch 01rd.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
= CDel branch hh.pngCDel split2.pngCDel node h.png
Uniform tiling 63-t0.png Uniform tiling 63-t01.png Uniform tiling 63-t1.png
Uniform tiling 333-t01.png
Uniform tiling 63-t12.png
Uniform tiling 333-t012.png
Uniform tiling 63-t2.png
Uniform tiling 333-t2.png
Uniform tiling 63-t02.png
Rhombitrihexagonal tiling snub edge coloring.png
Uniform tiling 63-t012.png Uniform tiling 63-snub.png Uniform tiling 333-t0.pngUniform tiling 333-t1.png Uniform tiling 333-t02.pngUniform tiling 333-t12.png Uniform tiling 63-h12.png
Uniform tiling 333-snub.png
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
CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 6.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node fh.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Uniform tiling 63-t2.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Rhombic star tiling.png Uniform tiling 63-t2.png Uniform tiling 63-t0.png Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg Uniform tiling 63-t0.png Rhombic star tiling.png Uniform tiling 63-t0.png


Triangle
symmetry
Extended
symmetry
Extended
diagram
Extended
order
Honeycomb diagrams
a1 [3[3] CDel node.pngCDel split1.pngCDel branch.png ×1 (None)
i2 <[3[3]>
= [6,3]
CDel node c1.pngCDel split1.pngCDel branch c2.png
= CDel node c1.pngCDel 3.pngCDel node c2.pngCDel 6.pngCDel node.png
×2 CDel node 1.pngCDel split1.pngCDel branch.png 1, CDel node.pngCDel split1.pngCDel branch 11.png 2
r6 [3[3[3]]
= [6,3]
CDel node c1.pngCDel split1.pngCDel branch c1.png
= CDel node c1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
×6 CDel node 1.pngCDel split1.pngCDel branch 11.png 3, CDel node h.pngCDel split1.pngCDel branch hh.png (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 CDel node 1.pngCDel split1.pngCDel branch.png CDel node 1.pngCDel split1.pngCDel branch 10l.png CDel node.pngCDel split1.pngCDel branch 10l.png CDel node.pngCDel split1.pngCDel branch 11.png CDel node.pngCDel split1.pngCDel branch 01l.png CDel node 1.pngCDel split1.pngCDel branch 01l.png CDel node 1.pngCDel split1.pngCDel branch 11.png CDel node h.pngCDel split1.pngCDel branch hh.png
Image
Vertex figure
Uniform tiling 333-t0.png
(3.3)3
Uniform tiling 333-t01.png
3.6.3.6
Uniform tiling 333-t1.png
(3.3)3
Uniform tiling 333-t12.png
3.6.3.6
Uniform tiling 333-t2.png
(3.3)3
Uniform tiling 333-t02.png
3.6.3.6
Uniform tiling 333-t012.png
6.6.6
Uniform tiling 333-snub.png
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)
Hexagonal tiling 4-colors.png Weaved hexagonal tiling.png

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.

Hexagonal tiling circle packing.svg

See also

References

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

External links








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