Hexagonal tiling

Short description: Regular tiling of a two-dimensional space

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} (as a truncated triangular tiling).

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

Hexagonal tiling is the densest way to arrange circles in two dimensions. The honeycomb conjecture states that 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, 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.

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

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  Chicken wire fencing

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  Graphene

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  A carbon nanotube can be seen as a hexagon tiling on a cylindrical surface

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  Hexagonal Persian tile c.1955

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 sphere packings in three dimensions. 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 chamfered 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.

The chamfered hexagonal tiling degenerates into a rhombille tiling at the limit

Hexagons (H)	Chamfered hexagons (cH)			Rhombi (daH)

Related tilings

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	Inset	Dual Tilings
Original		1/3 dissected	2/3 dissected	fully dissected		E to IH to FH to H

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 a 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 a 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 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 on the regular hexagonal tiling (or the dual triangular tiling).

Drawing the tiles colored red on the original faces, yellow at the original vertices, and blue along the original edges, there are 8 forms, 7 of 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. 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

1	2		3
p2, 2222	pgg, 22×	p2, 2222	p3, 333
	120px	120px
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. 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.

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 has 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). The gap inside each hexagon allows for one circle, creating the densest packing from the triangular tiling, with each circle in contact with a maximum of 6 circles.

Related regular complex apeirogons

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.

The first is made of 2-edges, three around every vertex, the 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

See also

• 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: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. https://archive.org/details/isbn_0716711931.  (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 [1]

External links

• Weisstein, Eric W.. "Hexagonal Grid". http://mathworld.wolfram.com/HexagonalGrid.html. 
  • Weisstein, Eric W.. "Regular tessellation". http://mathworld.wolfram.com/RegularTessellation.html. 
  • Weisstein, Eric W.. "Uniform tessellation". http://mathworld.wolfram.com/UniformTessellation.html. 
• Klitzing, Richard. "2D Euclidean tilings o3o6x – hexat – O3". https://bendwavy.org/klitzing/dimensions/flat.htm#2D. 

Fundamental convex regular and uniform honeycombs in dimensions 2-9
Space	Family	[math]\displaystyle{ {\tilde{A}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{C}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{B}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{D}}_{n-1} }[/math]	[math]\displaystyle{ {\tilde{G}}_2 }[/math] / [math]\displaystyle{ {\tilde{F}}_4 }[/math] / [math]\displaystyle{ {\tilde{E}}_{n-1} }[/math]
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