Hexagonal tiling

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Hexagonal tiling
Hexagonal tiling
Type Regular tiling
Vertex configuration 6.6.6 (or 63)
Hexagonal tiling vertfig.png
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

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.

Applications[edit]

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[edit]

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
Symmetry p6m, (*632) p3m1, (*333) p6m, (*632) p6, (632)
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)
Wythoff {6,3} t{3,6} t{3[3]}
Wythoff 3 | 6 2 2 6 | 3 3 3 3 |
Coxeter 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 H cH

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

Chamfered hexagonal tiling[edit]

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)
Uniform tiling 63-t0.png Chamfered hexagonal tiling.png Truncated rhombille tiling.png Chamfered hexagonal tiling2.png Rhombic star tiling.png

Related polyhedra and tilings[edit]

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

Regular tiling Dissection 2-uniform tilings Regular tiling
1-uniform 1.png
Original
Regular hexagon.svg
Triangular tiling vertfig.png
2-uniform 10.png
1/3 dissected
2-uniform 19.png
2/3 dissected
1-uniform 11.png
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 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

Symmetry mutations[edit]

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.

*n62 symmetry mutation of regular tilings: 6n or {6,n'}
Spherical Euclidean Hyperbolic tilings
Hexagonal dihedron.png
{6,2}
Uniform tiling 63-t0.png
{6,3}
H2 tiling 246-1.png
{6,4}
H2 tiling 256-1.png
{6,5}
H2 tiling 266-4.png
{6,6}
H2 tiling 267-4.png
{6,7}
H2 tiling 268-4.png
{6,8}
... H2 tiling 26i-4.png
{6,∞}

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

*n32 symmetry mutation of regular tilings: n3 or {n',3}
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
Spherical trigonal hosohedron.png Uniform tiling 332-t0-1-.png Uniform tiling 432-t0.png Uniform tiling 532-t0.png Uniform polyhedron-63-t0.png H2 tiling 237-1.png H2 tiling 238-1.png H2 tiling 23i-1.png H2 tiling 23j12-1.png H2 tiling 23j9-1.png H2 tiling 23j6-1.png H2 tiling 23j3-1.png
{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 hyperb. 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
Hexagonal dihedron.png Uniform tiling 332-t12.png Uniform tiling 432-t12.png Uniform tiling 532-t12.png Uniform tiling 63-t12.png H2 tiling 237-6.png H2 tiling 238-6.png H2 tiling 23i-6.png H2 tiling 23j12-6.png H2 tiling 23j9-6.png H2 tiling 23j-6.png
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
n-kis
figures
Hexagonal Hosohedron.svg Spherical triakis tetrahedron.png Spherical tetrakis hexahedron.png Spherical pentakis dodecahedron.png Uniform tiling 63-t2.png Order3 heptakis heptagonal til.png H2checkers 334.png H2checkers 33i.png
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 n-gons at the truncated vertices, and nonregular hexagonal faces.

Symmetry mutations of dual quasiregular tilings: V(3.n)2
Spherical Euclidean Hyperbolic
*n32 *332 *432 *532 *632 *732 *832... *∞32
Tiling Uniform tiling 432-t0.png Spherical rhombic dodecahedron.png Spherical rhombic triacontahedron.png Rhombic star tiling.png Order73 qreg rhombic til.png Uniform dual tiling 433-t01-yellow.png Ord3infin qreg rhombic til.png
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[edit]

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}
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
Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg Uniform tiling 63-t0.png Uniform tiling 63-t01.png Uniform tiling 63-t1.png Uniform tiling 63-t12.png Uniform tiling 63-t2.png Uniform tiling 63-t02.png Uniform tiling 63-t012.png Uniform tiling 63-snub.png
Config. 63 3.12.12 (6.3)2 6.6.6 36 3.4.6.4 4.6.12 3.3.3.3.6

Topologically equivalent tilings[edit]

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]

The 2-uniform and 3-uniform 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 non-edge-to-edge tiling of hexagons and larger triangles.[2]

It can also be distorted into a chiral 4-colored tri-directional 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)
Uniform tiling 63-t12.png Gyrated hexagonal tiling2.png Truncated rhombille tiling.png Weaved hexagonal tiling2.png
p3m1, (*333) p3, (333) p6m (*632) p2 (2222)
Uniform tiling 333-t012.png Gyrated hexagonal tiling1.png Hexagonal tiling 4-colors.png Weaved hexagonal tiling.png

Circle packing[edit]

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).[3] 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.

Hexagonal tiling circle packing.png Hexagonal tiling circle packing2.png

See also[edit]

References[edit]

  1. ^ Tilings and Patterns, from list of 107 isohedral tilings, p.473-481
  2. ^ Tilings and patterns, uniform tilings that are not edge-to-edge
  3. ^ Order in Space: A design source book, Keith Critchlow, p.74-75, pattern 2

External links[edit]