Hexagonal tiling

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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	face-transitive

In

English mathematician John Conway called it a hextille.

The

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

  cylindrical

  surface
• 
  Hexagonal Persian tile c.1955

The hexagonal tiling appears in many crystals. In three dimensions, the

, amongst other materials, forms a face-centered cubic lattice.

Uniform colorings

There are three distinct

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

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

Related tilings

The hexagons can be dissected into sets of 6 triangles. This process leads to two

:

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

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

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
	*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
	*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 (××)		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 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

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

Wikimedia Commons has media related to Order-3 hexagonal tiling.

• 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

• ISBN 0-486-61480-8
  p. 296, Table II: Regular honeycombs
• ISBN 0-7167-1193-1
  . (Chapter 2.1: Regular and uniform tilings, pp. 58–65)
• ISBN 0-486-23729-X
  .
• John H. Conway, Heidi Burgiel,
  ISBN 978-1-56881-220-5 [1]

• 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".

regular and uniform honeycombs in dimensions 2–9
Space	Family	${\displaystyle {\tilde {A}}_{n-1}}$	${\displaystyle {\tilde {C}}_{n-1}}$	${\displaystyle {\tilde {B}}_{n-1}}$	${\displaystyle {\tilde {D}}_{n-1}}$	${\displaystyle {\tilde {G}}_{2}}$ / ${\displaystyle {\tilde {F}}_{4}}$ / ${\displaystyle {\tilde {E}}_{n-1}}$
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
E10	Uniform 10-honeycomb	{3[11]}	δ11	hδ11	qδ11
En-1	Uniform (n-1)-honeycomb	{3[n]}	δn	hδn	qδn	1k2 • 2k1 • k21