Hexagonal tiling

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).

Hexagonal tiling

TypeRegular tiling
Vertex configuration6.6.6 (or 63)
Face configurationV3.3.3.3.3.3 (or V36)
Schläfli symbol(s){6,3}
t{3,6}
Wythoff symbol(s)3 | 6 2
2 6 | 3
3 3 3 |
Coxeter diagram(s)

Symmetryp6m, [6,3], (*632)
Rotation symmetryp6, [6,3]+, (632)
DualTriangular tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

English mathematician John Conway called 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 honeycomb (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.

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 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 three 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. The same counting is used in the Goldberg polyhedra, with a notation {p+,3}h,k, and can be applied to hyperbolic tilings for p>6.

k-uniform 1-uniform 2-uniform 3-uniform
Symmetry p6m, (*632) p3m1, (*333) p6m, (*632) p6, (632)
Picture
Colors 1 2 3 2 4 2 7
(h,k) (1,0) (1,1) (2,0) (2,1)
Schläfli {6,3} t{3,6} t{3[3]}
Wythoff 3 | 6 2 2 6 | 3 3 3 3 |
Coxeter
Conway H tΔ cH=t6daH wH=t6dsH

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

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

Original

1/3 dissected
2/3 dissected
fully dissected
Regular Tiling Inset 2-Uniform Duals Regular Tiling

Original
1/3 inset
2/3 inset
fully inset

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

It is also possible to subdivide the prototiles of certain hexagonal tilings by two, three, four or nine equal pentagons:


Pentagonal tiling type 1 with overlays of regular hexagons (each comprising 2 pentagons).
pentagonal tiling type 3 with overlays of regular hexagons (each comprising 3 pentagons).
Pentagonal tiling type 4 with overlays of semiregular hexagons (each comprising 4 pentagons).
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.

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

It is similarly related to the uniform truncated polyhedra with vertex figure n.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.

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.)

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 2-isohedral keeping chiral pairs distinct.

3 types of monohedral convex hexagonal tilings
123
p2, 2222pgg, 22×p2, 2222p3, 333

b=e
B+C+D=360°
b=e, d=f
B+C+E=360°
a=f, b=c, d=e
B=D=F=120°

2-tile lattice
4-tile lattice
3-tile 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] Single-color (1-tile) lattices are parallelogon hexagons.

13 isohedrally-tiled hexagons
pg (××)p2 (2222)p3 (333)pmg (22*)
pgg (22×)p31m (3*3)p2 (2222)cmm (2*22)p6m (*632)

Other isohedrally-tiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edge-to-edge, but interpreted as colinear adjacent edges:

Isohedrally-tiled quadrilaterals
pmg (22*)pgg (22×)cmm (2*22)p2 (2222)

Parallelogram
Trapezoid
Parallelogram
Rectangle
Parallelogram
Rectangle
Rectangle
Isohedrally-tiled pentagons
p2 (2222)pgg (22×)p3 (333)

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.[3]

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. A chevron pattern has pmg (22*) symmetry, which is lowered to p1 (°) with 3 or 4 colored tiles.

Regular Gyrated Regular Weaved Chevron
p6m, (*632) p6, (632) p6m (*632) p6 (632) p1 (°)
p3m1, (*333) p3, (333) p6m (*632) p2 (2222) p1 (°)

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 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.

There are 2 regular complex apeirogons, sharing the vertices of the hexagonal tiling. Regular complex apeirogons have vertices and edges, where edges can contain 2 or more vertices. Regular apeirogons p{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices, and vertex figures are r-gonal.[5]

The first is made of 2-edges, three around every vertex, second has hexagonal edges, three around every vertex. A third complex apeirogon, sharing the same vertices, is quasiregular, which alternates 2-edges and 6-edges.

2{12}3 or 6{4}3 or
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See also

References
  1. Tilings and Patterns, Sec. 9.3 Other Monohedral tilings by convex polygons
  2. Tilings and Patterns, from list of 107 isohedral tilings, pp. 473–481
  3. Tilings and patterns, uniform tilings that are not edge-to-edge
  4. Order in Space: A design source book, Keith Critchlow, pp. 74–75, pattern 2
  5. Coxeter, Regular Complex Polytopes, pp. 111-112, p. 136.
  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 p. 296, Table II: Regular honeycombs
  • Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, pp. 58–65)
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 35. ISBN 0-486-23729-X.
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space Family / /
E2 Uniform tiling {3[3]} δ3 hδ3 qδ3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 hδ4 qδ4
E4 Uniform 4-honeycomb {3[5]} δ5 hδ5 qδ5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 hδ6 qδ6
E6 Uniform 6-honeycomb {3[7]} δ7 hδ7 qδ7 222
E7 Uniform 7-honeycomb {3[8]} δ8 hδ8 qδ8 133 • 331
E8 Uniform 8-honeycomb {3[9]} δ9 hδ9 qδ9 152 • 251 • 521
E9 Uniform 9-honeycomb {3[10]} δ10 hδ10 qδ10
En-1 Uniform (n-1)-honeycomb {3[n]} δn hδn qδn 1k2 • 2k1 • k21
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