Hexagonal tiling
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Hexagonal tiling  

Type  Regular tiling 
Vertex configuration  6.6.6 (or 6^{3}) 
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  Vertextransitive, edgetransitive, facetransitive 
6.6.6 (or 6^{3}) 
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.
Contents
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 threedimensional structure for making beehives (or rather, soap bubbles) was investigated by Lord Kelvin, who believed that the Kelvin structure (or bodycentered cubic lattice) is optimal. However, the less regular WeairePhelan 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

A carbon nanotube can be seen as a hexagon tiling on a cylindrical surface
The hexagonal tiling appears in many crystals. In three dimensions, the facecentered 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 facecentered cubic being the more regular of the two. Pure copper, amongst other materials, forms a facecentered 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.
kuniform  1uniform  2uniform  3uniform  

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]^{+} 

CoxeterDynkin diagram  
Conway polyhedron notation  H  tH  teH  t6daH  t6dateH 
The 3color tiling is a tessellation generated by the order3 permutohedrons.
Related polyhedra and tilings
This tiling is topologically related to regular polyhedra with vertex figure n^{3}, 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.
Symmetry *n42 [n,3] 
Spherical  Euclidean  Compact hyperbolic  Paracompact  

*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{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  
nkis 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 ngons 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 rhombohexagonal 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.)
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} s_{2}{6,3} 
tr{6,3}  sr{6,3}  h{6,3} {3^{[3]}} 
h_{2}{6,3} r{3^{[3]}} 
s{3,6} s{3^{[3]}} 
= 
= 
= 
= or 
= or 
= 

Uniform duals  
V6^{3}  V3.12^{2}  V(3.6)^{2}  V6^{3}  V3^{6}  V3.4.12.4  V.4.6.12  V3^{4}.6  V3^{6}  V(3.6)^{2}  V3^{6} 
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 (facetransitivity) and vertextransitivity, there are 12 variations, with the first 7 identified as quadrilaterals that don't connect edgetoedge, or as hexagons with two pairs of colinear edges. Symmetry given assumes all faces are the same color.^{1}

Parallelogram
p2 symmetry 
rectangle
pgg symmetry 
Trapezoid
pmg symmetry 
Regular hexagon
p6m symmetry
It can also be distorted into a chiral 4colored tridirectional weaved pattern, distorting some hexagons into parallelograms. The weaved pattern with 4colored faces have rotational 632 (p6) symmetry.
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.
See also
Wikimedia Commons has media related to Order3 hexagonal tiling. 
 Hexagonal lattice
 Hexagonal prismatic honeycomb
 Tilings of regular polygons
 List of uniform tilings
 List of regular polytopes
 Hexagonal tiling honeycomb
 Hex map board game design
References
 ^ Tilings and Patterns, from list of 107 isohedral tilings, p.473481
 Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0486614808 p. 296, Table II: Regular honeycombs
 Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0716711931. (Chapter 2.1: Regular and uniform tilings, p. 5865)
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 35. ISBN 048623729X.
 John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, The Symmetries of Things 2008, ISBN 9781568812205 [1]
External links
 Weisstein, Eric W., "Hexagonal Grid", MathWorld.
 Richard Klitzing, 2D Euclidean tilings, o3o6x  hexat  O3
Fundamental convex regular and uniform honeycombs in dimensions 2–11  

Family  / /  
Uniform tiling  {3^{[3]}}  δ_{3}  hδ_{3}  qδ_{3}  Hexagonal 
Uniform convex honeycomb  {3^{[4]}}  δ_{4}  hδ_{4}  qδ_{4}  
Uniform 5honeycomb  {3^{[5]}}  δ_{5}  hδ_{5}  qδ_{5}  24cell honeycomb 
Uniform 6honeycomb  {3^{[6]}}  δ_{6}  hδ_{6}  qδ_{6}  
Uniform 7honeycomb  {3^{[7]}}  δ_{7}  hδ_{7}  qδ_{7}  2_{22} 
Uniform 8honeycomb  {3^{[8]}}  δ_{8}  hδ_{8}  qδ_{8}  1_{33} • 3_{31} 
Uniform 9honeycomb  {3^{[9]}}  δ_{9}  hδ_{9}  qδ_{9}  1_{52} • 2_{51} • 5_{21} 
Uniform nhoneycomb  {3^{[n]}}  δ_{n}  hδ_{n}  qδ_{n}  1_{k2} • 2_{k1} • k_{21} 
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