Hexagonal tiling
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Hexagonal tiling  

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 or hexagonal tessellation 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 honeycomb (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.
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. Silicene is similar.
Chicken wire consists of a hexagonal lattice (often not regular) of wires.

The densest circle packing is arranged like the hexagons in this tiling

Chicken Wire closeup.jpg
Chicken wire fencing

Graphene xyz.jpg

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  

Symmetry  p6m, (*632)  p3m1, (*333)  p6m, (*632)  p6, (632)  
Picture  100px  100px  100px  
Colors  1  2  3  2  4  2  7 
(h,k)  (1,0)  (1,1)  (2,0)  (2,1)  
Wythoff  {6,3}  t{3,6}  t{3^{[3]}}  
Wythoff  3  6 2  2 6  3  3 3 3   
Coxeter  
Conway  H  tΔ  cH 
The 3color tiling is a tessellation generated by the order3 permutohedrons.
Chamfered hexagonal tiling
A chamferred hexagonal tiling replacing edges with new hexagons and transforms into another hexagonal tiling. In the limit, the original faces disappear, and the new hexagons degenerate into rhombi, and it becomes a rhombic tiling.
Hexagons (H)  Chamfered hexagons (cH)  Rhombi (daH)  

Related tilings
The hexagons can be dissected into sets of 6 triangles. This process leads to two 2uniform tilings, and the triangular tiling:
Regular tiling  Dissection  2uniform tilings  Regular tiling  

120px Original 
1/3 dissected 
120px 2/3 dissected 
120px fully dissected 
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.
160px Rhombic tiling 
Hexagonal tiling 
160px Fencing uses this relation 
It is also possible to subdivide the prototiles of certain hexagonal tilings by two, three, four or nine equal pentagons:
175px Pentagonal tiling type 1 with overlays of regular hexagons (each comprising 2 pentagons). 
175px pentagonal tiling type 3 with overlays of regular hexagons (each comprising 3 pentagons). 
175px Pentagonal tiling type 4 with overlays of semiregular hexagons (each comprising 4 pentagons). 
175px Pentagonal tiling type 3 with overlays of two sizes of regular hexagons (comprising 3 and 9 pentagons respectively). 
Symmetry mutations
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.
*n62 symmetry mutation of regular tilings: {6,n}  

Spherical  Euclidean  Hyperbolic tilings  
{6,2} 
{6,3} 
{6,4} 
{6,5} 
{6,6} 
{6,7} 
{6,8} 
...  {6,∞} 
This tiling is topologically related to regular polyhedra with vertex figure n^{3}, as a part of sequence that continues into the hyperbolic plane.
*n32 symmetry mutation of regular tilings: {n,3}  

Spherical  Euclidean  Compact hyperb.  Paraco.  Noncompact hyperbolic  
{2,3}  {3,3}  {4,3}  {5,3}  {6,3}  {7,3}  {8,3}  {∞,3}  {12i,3}  {9i,3}  {6i,3}  {3i,3} 
It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6.
*n32 symmetry mutation of truncated tilings: n.6.6  

Sym. *n42 [n,3] 
Spherical  Euclid.  Compact  Parac.  Noncompact hyperbolic  
*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3] 
[12i,3]  [9i,3]  [6i,3]  
Truncated figures 

Config.  2.6.6  3.6.6  4.6.6  5.6.6  6.6.6  7.6.6  8.6.6  ∞.6.6  12i.6.6  9i.6.6  6i.6.6  
nkis figures 

Config.  V2.6.6  V3.6.6  V4.6.6  V5.6.6  V6.6.6  V7.6.6  V8.6.6  V∞.6.6  V12i.6.6  V9i.6.6  V6i.6.6 
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.
Symmetry mutations of dual quasiregular tilings: V(3.n)^{2}  

*n32  Spherical  Euclidean  Hyperbolic  
*332  *432  *532  *632  *732  *832...  *∞32  
Tiling  
Conf.  V(3.3)^{2}  V(3.4)^{2}  V(3.5)^{2}  V(3.6)^{2}  V(3.7)^{2}  V(3.8)^{2}  V(3.∞)^{2} 
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  

Fundamental domains 
Symmetry: [6,3], (*632)  [6,3]^{+}, (632)  
{6,3}  t{6,3}  r{6,3}  t{3,6}  {3,6}  rr{6,3}  tr{6,3}  sr{6,3}  
Config.  6^{3}  3.12.12  (6.3)^{2}  6.6.6  3^{6}  3.4.6.4  4.6.12  3.3.3.3.6 
Monohedral convex hexagonal tilings
There are 3 types of monohedral convex hexagonal tilings.^{[1]} They are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains glide reflections, and is 2isohedral keeping chiral pairs distinct.
1  2  3  

p2, 2222  pgg, 22×  p2, 2222  p3, 333 
120px  120px  120px  120px 
120px b=e B+C+D=360° 
120px b=e, d=f B+C+E=360° 
120px a=f, b=c, d=e B=D=F=120° 

120px 2tile lattice 
120px 4tile lattice 
120px 3tile lattice 
Topologically equivalent tilings
Hexagonal tilings can be made with the identical {6,3} topology as the regular tiling (3 hexagons around every vertex). With isohedral faces, there are 13 variations. Symmetry given assumes all faces are the same color. Colors here represent the lattice positions.^{[2]} Singlecolor (1tile) lattices are parallelogon hexagons.
pg (××)  p2 (2222)  p3 (333)  pmg (22*)  

100px  100px  100px  
pgg (22×)  p31m (3*3)  p2 (2222)  cmm (2*22)  p6m (*632)  
100px  100px 
Other isohedrallytiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edgetoedge, but interpreted as colinear adjacent edges:
pmg (22*)  pgg (22×)  cmm (2*22)  p2 (2222)  

100px Parallelogram 
100px Trapezoid 
Parallelogram 
Rectangle 
100px Parallelogram 
100px Rectangle 
100px Rectangle 
p2 (2222)  pgg (22×)  p3 (333) 

120px  120px  120px 
The 2uniform and 3uniform tessellations have a rotational degree of freedom which distorts 2/3 of the hexagons, including a colinear case that can also be seen as a nonedgetoedge tiling of hexagons and larger triangles.^{[3]}
It can also be distorted into a chiral 4colored tridirectional weaved pattern, distorting some hexagons into parallelograms. The weaved pattern with 2 colored faces have rotational 632 (p6) symmetry.
Regular  Gyrated  Regular  Weaved 

p6m, (*632)  p6, (632)  p6m (*632)  p6 (632) 
150px  150px  
p3m1, (*333)  p3, (333)  p6m (*632)  p2 (2222) 
150px  150px  150px 
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).^{[4]} The lattice volume is filled by two circles, so the circles can be alternately colored. The gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, 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
 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.CS1 maint: multiple names: authors list (link)<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles> (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.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>
 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 2–10  

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} 