| Hexagonal tiling | |
|---|---|
| Type | Regular tiling |
| Vertex configuration | 6.6.6 (or 63) |
| Schläfli symbol(s) | {6,3} t0,1{3,6} |
| Wythoff symbol(s) | 3 | 6 2 2 6 | 3 3 3 3 | |
| Coxeter-Dynkin(s) | |
| Symmetry | p6m, [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.
Contents |
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.
There are 3 distinct uniform colorings of a hexagonal tiling, all generated from reflective symmetry of Wythoff constructions.
| k-uniform | 1-uniform | 2-uniform | |||
|---|---|---|---|---|---|
| Picture | |||||
| Schläfli symbol | {6,3} | t{3,6} | t0,1,2{3[3]} | ||
| Wythoff symbol | 3 | 6 2 | 2 6 | 3 | 3 3 3 | | ||
| symmetry | *632 (p6m) | *632 (p6m) | *333 (p3) | *632 (p6m) | *632 (p6m) |
| Coxeter-Dynkin diagram | |||||
| Conway polyhedron notation | H | tH | teH | t6daH | t6dateH |
The 3-color tiling is a tessellation generated by the order-3 permutohedrons.
The hexagonal tiling can be stretched and adjusted to other geometric proportions and different symmetries.
The standard brick pattern can be considered a nonregular hexagonal tiling. Each rectangular brick has vertices inserted on the two long edges, dividing them into two collinear edges.
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. The herringbone pattern is also a distorted hexagonal tiling.
| Regular hexagons | Hexagonal weave |
|---|---|
| p6m (*632) | p6 (632) |
| Brick pattern | Herringbone |
| p4g (4*2) | |
| | | |
This tiling is topologically related to regular polyhedra with vertex figure n3, as a part of sequence that continues into the hyperbolic plane.
(33) |
(43) |
(53) |
(63) tiling |
(73) tiling |
It is similarly related to the uniform truncated polyhedra with vertex figure n.6.6.
(3.6.6) |
(4.6.6) |
(5.6.6) |
(6.6.6) tiling |
(7.6.6) tiling |
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 |
||
Alternate truncated cube |
Truncated rhombic dodecahedron |
Truncated rhombic triacontahedron |
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 |
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.)
| Tiling | Schläfli symbol |
Wythoff symbol |
Vertex figure |
Image |
|---|---|---|---|---|
| Hexagonal tiling | t0{6,3} | 3 | 6 2 | 63 | |
| Truncated hexagonal tiling | t0,1{6,3} | 2 3 | 6 | 3.12.12 | |
| Rectified hexagonal tiling (Trihexagonal tiling) |
t1{6,3} | 2 | 6 3 | (3.6)2 | |
| Bitruncated hexagonal tiling (Truncated triangular tiling) |
t1,2{6,3} | 2 6 | 3 | 6.6.6 | |
| Dual hexagonal tiling (Triangular tiling) |
t2{6,3} | 6 | 3 2 | 36 | |
| Cantellated hexagonal tiling (Rhombitrihexagonal tiling) |
t0,2{6,3} | 6 3 | 2 | 3.4.6.4 | |
| Omnitruncated hexagonal tiling (Truncated trihexagonal tiling) |
t0,1,2{6,3} | 6 3 2 | | 4.6.12 | |
| Snub hexagonal tiling | s{6,3} | | 6 3 2 | 3.3.3.3.6 |