Hexagonal tiling

Hexagonal tiling

Type	Regular tiling
Vertex configuration	6.6.6 (or 6³)
Face configuration	V3.3.3.3.3.3 (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)
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 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.

The densest circle packing is arranged like the hexagons in this tiling
Chicken wire fencing
Graphene
A carbon nanotube can be seen as a hexagon tiling on a cylindrical surface
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 replaces 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 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.

*n62 symmetry mutation of regular tilings: {6,n} v t e
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 n³, as a part of a sequence that continues into the hyperbolic plane.

*n32 symmetry mutation of regular tilings: {n,3} v t e
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 v t e
Sym. *n42 [n,3]	Spherical				Euclid.	Compact		Parac.	Noncompact hyperbolic
Sym. *n42 [n,3]	*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
n-kis 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 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
*n32	Spherical			Euclidean	Hyperbolic
*n32	*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 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.)

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.	6³	3.12.12	(6.3)²	6.6.6	3⁶	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 2-isohedral keeping chiral pairs distinct.

3 types of monohedral convex hexagonal tilings
1	2		3
p2, 2222	pgg, 22×	p2, 2222	p3, 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 (××)	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 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).^[4] 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.^[5]

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

References

^ Tilings and patterns, Sec. 9.3 Other Monohedral tilings by convex polygons
^ Tilings and patterns, from list of 107 isohedral tilings, pp. 473–481
^ Tilings and patterns, uniform tilings that are not edge-to-edge
^ Order in Space: A design source book, Keith Critchlow, pp. 74–75, pattern 2
^ 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 [1]

External links

Weisstein, Eric W. "Hexagonal Grid". MathWorld.
- Weisstein, Eric W. "Regular tessellation". MathWorld.
- Weisstein, Eric W. "Uniform tessellation". MathWorld.
Klitzing, Richard. "2D Euclidean tilings o3o6x – hexat – O3".

v t e Fundamental convex regular and uniform honeycombs in dimensions 2–9
Space	Family	${\tilde {A}}_{n-1}$	${\tilde {C}}_{n-1}$	${\tilde {B}}_{n-1}$	${\tilde {D}}_{n-1}$	${\tilde {G}}_{2}$ / ${\tilde {F}}_{4}$ / ${\tilde {E}}_{n-1}$
E²	Uniform tiling	0_[3]	δ₃	hδ₃	qδ₃	Hexagonal
E³	Uniform convex honeycomb	0_[4]	δ₄	hδ₄	qδ₄
E⁴	Uniform 4-honeycomb	0_[5]	δ₅	hδ₅	qδ₅	24-cell honeycomb
E⁵	Uniform 5-honeycomb	0_[6]	δ₆	hδ₆	qδ₆
E⁶	Uniform 6-honeycomb	0_[7]	δ₇	hδ₇	qδ₇	2₂₂
E⁷	Uniform 7-honeycomb	0_[8]	δ₈	hδ₈	qδ₈	1₃₃ • 3₃₁
E⁸	Uniform 8-honeycomb	0_[9]	δ₉	hδ₉	qδ₉	1₅₂ • 2₅₁ • 5₂₁
E⁹	Uniform 9-honeycomb	0_[10]	δ₁₀	hδ₁₀	qδ₁₀
E¹⁰	Uniform 10-honeycomb	0_[11]	δ₁₁	hδ₁₁	qδ₁₁
E^n-1	Uniform (n-1)-honeycomb	0_[n]	δ_n	hδ_n	qδ_n	1_k2 • 2_k1 • k₂₁

[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

[Critchlow-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.

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