Hexagonal tiling

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Hexagonal tiling
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Typeregular tiling
Tileregular hexagon
Vertex configuration6.6.6
Wallpaper groupp6m
Dualtriangular tiling
Propertiesvertex-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.

Structure and properties

The hexagonal tiling has a structure consisting of a regular hexagon only as its prototile, sharing two vertices with other identical ones, an example of monohedral tiling.[1] Each vertex at the tiling is surrounded by three regular hexagons, denoted as 6.6.6 by vertex configuration.[2] The dual of a hexagonal tiling is triangular tiling, because the center of each hexagonal tiling connects to another center of one, forming equilateral triangles.[3]

Every mutually incident vertex, edge, and tile of a hexagonal tiling can be act transitively to another of those three by mapping the first ones to the second through the symmetry operation. In other words, they are vertex-transitive (mapping the vertex of a tile to another), edge-transitive (mapping the edge to another), and face-transitive (mapping regular hexagonal tile to another). From these, the hexagonal tiling is categorized as one of three regular tilings; the remaining being its dual and square tiling.Lua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found. The symmetry group of a hexagonal tiling is p6m.Lua error in package.lua at line 80: module 'Module:Footnotes/anchor_id_list' not found.

Applications

File:Kissing-2d.svg
The densest circle packing is arranged like the hexagons in this tiling

If a circle is inscribed in each hexagon, the resulting figure is the densest way to arrange circles in two dimensions; its packing density is π23≈0.907.[4] The honeycomb theorem states that hexagonal tiling is the best way to divide a surface into regions of equal area with the least total perimeter.[5] 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.

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File:Graphene xyz.jpg
Graphene
File:Chicken Wire close-up.jpg
Chicken wire fencing

The hexagonal tiling is commonly found in nature, such as the sheet of graphene with strong covalent carbon bonds. Tubular graphene sheets have been synthesised, known as carbon nanotubes.[6] They have many potential applications, due to their high tensile strength and electrical properties. Silicene has a similar structure as graphene.

Chicken wire consists of a hexagonal lattice of wires, although the shape is not regular.[7]

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 File:Uniform tiling 63-t0.svg File:Uniform tiling 63-t12.svg File:Uniform tiling 333-t012.svg File:Truncated rhombille tiling.svg File:Hexagonal tiling 4-colors.svg File:Hexagonal tiling 2-1.svg File:Hexagonal tiling 7-colors.svg
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 Template:CDD Template:CDD Template:CDD
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.

File:ChamferedHexTilingAnimation.gif
The chamfered hexagonal tiling degenerates into a rhombille tiling at the limit
Hexagons (H) Chamfered hexagons (cH) Rhombi (daH)
File:Uniform tiling 63-t0.svg File:Chamfered hexagonal tiling.svg File:Truncated rhombille tiling.svg File:Chamfered hexagonal tiling2.svg File:Rhombic star tiling.svg

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
File:1-uniform n1.svg
Original 		File:Regular hexagon.svgFile:Vertex type 3-3-3-3-3-3.svg File:2-uniform n10.svg
1/3 dissected 		File:2-uniform n19.svg
2/3 dissected 		File:1-uniform n11.svg
fully dissected 		File:Inset Variations of Dual Uniform Tiling.svg File:E to IH to FH to H Insets.gif
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.

File:Kah 3 6 romb.svg
Rhombic tiling 		File:Uniform tiling 63-t0.svg
Hexagonal tiling 		File:Valle del Chubut (5).JPG
Fencing uses this relation

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

File:Pent-Hex-Type1-2.svg
Pentagonal tiling type 1 with overlays of regular hexagons (each comprising 2 pentagons). 		File:Pent-Hex-Type3-3.svg
pentagonal tiling type 3 with overlays of regular hexagons (each comprising 3 pentagons). 		File:Pent-Hex-Type4-4.svg
Pentagonal tiling type 4 with overlays of semiregular hexagons (each comprising 4 pentagons). 		File:Pent-Hex-Type3-9.svg
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 Template:CDD, progressing to infinity.

*n62 symmetry mutation of regular tilings: {6,n} Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
Spherical Euclidean Hyperbolic tilings
File:Hexagonal dihedron.svg
{6,2} 		File:Uniform tiling 63-t0.svg
{6,3} 		File:H2 tiling 246-1.png
{6,4} 		File:H2 tiling 256-1.png
{6,5} 		File:H2 tiling 266-4.png
{6,6} 		File:H2 tiling 267-4.png
{6,7} 		File:H2 tiling 268-4.png
{6,8} 		... File:H2 tiling 26i-4.png
{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} Lua error in package.lua at line 80: module 'Module:Navbar/configuration' not found.
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
File:Spherical trigonal hosohedron.svg File:Uniform tiling 332-t0-1-.svg File:Uniform tiling 432-t0.svg File:Uniform tiling 532-t0.svg File:Uniform polyhedron-63-t0.png File:Heptagonal tiling.svg File:H2-8-3-dual.svg File:H2-I-3-dual.svg File:H2 tiling 23j12-1.png File:H2 tiling 23j9-1.png File:H2 tiling 23j6-1.png File:H2 tiling 23j3-1.png
{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.

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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 		File:Hexagonal dihedron.svg File:Uniform tiling 332-t12.svg File:Uniform tiling 432-t12.svg File:Uniform tiling 532-t12.png File:Uniform tiling 63-t12.svg File:Truncated order-7 triangular tiling.svg File:H2-8-3-trunc-primal.svg File:H2 tiling 23i-6.png File:H2 tiling 23j12-6.png File:H2 tiling 23j9-6.png File:H2 tiling 23j-6.png
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 		File:Hexagonal Hosohedron.svg File:Spherical triakis tetrahedron.svg File:Spherical tetrakis hexahedron.svg File:Spherical pentakis dodecahedron.png File:Uniform tiling 63-t2.svg File:Heptakis heptagonal tiling.svg File:H2-8-3-kis-dual.svg File:H2checkers 33i.png
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 File:Uniform tiling 432-t0.svg File:Spherical rhombic dodecahedron.svg File:Spherical rhombic triacontahedron.svg File:Rhombic star tiling.svg File:7-3 rhombille tiling.svg File:H2-8-3-rhombic.svg File:Ord3infin qreg rhombic til.png
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

Monohedral convex hexagonal tilings

There are 3 types of monohedral convex hexagonal tilings.[8] 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
File:P6-type1.svg File:P6-type2.svg File:P6-type2-chiral coloring.svg File:P6-type3.svg
File:Prototile p6-type1.png
b = e
B + C + D = 360° 		File:Prototile p6-type2.png
b = e, d = f
B + C + E = 360° 		File:Prototile p6-type3.png
a = f, b = c, d = e
B = D = F = 120°
File:Lattice p6-type1.png
2-tile lattice 		File:Lattice p6-type2.png
4-tile lattice 		File:Lattice p6-type3.png
3-tile lattice

There are also 15 monohedral convex pentagonal tilings, as well as all quadrilaterals and triangles.

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.[9] Single-color (1-tile) lattices are parallelogon hexagons.

13 isohedrally-tiled hexagons
pg (××) p2 (2222) p3 (333) pmg (22*)
File:Isohedral tiling p6-1.svg File:Isohedral tiling p6-2.svg File:Isohedral tiling p6-3.svg File:Isohedral tiling p6-6.svg File:Isohedral tiling p6-9.svg File:Isohedral tiling p6-10.svg
pgg (22×) p31m (3*3) p2 (2222) cmm (2*22) p6m (*632)
File:Isohedral tiling p6-4.svg File:Isohedral tiling p6-5.png File:Isohedral tiling p6-8.svg File:Isohedral tiling p6-11.svg File:Isohedral tiling p6-7.svg File:Isohedral tiling p6-12.svg File:Isohedral tiling p6-13.svg

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)
File:Isohedral tiling p4-18.png
Parallelogram 		File:Isohedral tiling p4-20.png
Trapezoid 		File:Isohedral tiling p4-19.png
Parallelogram 		File:Isohedral tiling p4-19b.png
Rectangle 		File:Isohedral tiling p4-17.svg
Parallelogram 		File:Isohedral tiling p4-21.svg
Rectangle 		File:Isohedral tiling p4-22.svg
Rectangle
Isohedrally-tiled pentagons
p2 (2222) pgg (22×) p3 (333)
File:P5-type1.svg File:P5-type2.svg File:P5-type3.svg

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.[10]

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 (°)
File:Uniform tiling 63-t12.svg File:Gyrated hexagonal tiling2.svg File:Truncated rhombille tiling.svg File:Weaved hexagonal tiling2.svg File:Chevron hexagonal tiling-3-color.svg
p3m1, (*333) p3, (333) p6m (*632) p2 (2222) p1 (°)
File:Uniform tiling 333-t012.svg File:Gyrated hexagonal tiling1.svg File:Hexagonal tiling 4-colors.svg File:Weaved hexagonal tiling.svg File:Chevron hexagonal tiling-4-color.svg

Circle packing

File:Comparison hex tiling.svg
Hexagonal (mid­dle) vs equivalent circle (top) vs staggered rectan­gular tilings (bottom)

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).[11] 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.

File:1-uniform-1-circlepack.svg

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.[12]

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.

File:Complex apeirogon 2-12-3.svg File:Complex apeirogon 6-4-3.svg File:Truncated complex polygon 6-6-2.svg
2{12}3 or Template:CDD 6{4}3 or Template:CDD Template:CDD

See also

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References

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  1. ^ Page Module:Citation/CS1/styles.css has no content.Adams, Colin (2022). The Tiling Book: An Introduction to the Mathematical Theory of Tilings. American Mathematical Society. pp. 23. ISBN 9781470468972.
  2. ^ Page Module:Citation/CS1/styles.css has no content.Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns. W. H. Freeman. p. 21.
  3. ^ Page Module:Citation/CS1/styles.css has no content.Nelson, Roice; Segerman, Henry (2017). "Visualizing hyperbolic honeycombs". Journal of Mathematics and the Arts. 11 (1): 4–39. arXiv:1511.02851. doi:10.1080/17513472.2016.1263789.
  4. ^ Page Module:Citation/CS1/styles.css has no content.Chang, Hai-Chau; Wang, Lih-Chung (22 September 2010). "A Simple Proof of Thue's Theorem on Circle Packing". arXiv:1009.4322 [math.MG].
  5. ^ Page Module:Citation/CS1/styles.css has no content.Hales, Thomas C. (January 2001). "The Honeycomb Conjecture". Discrete and Computational Geometry. 25 (1): 1–22. arXiv:math/9906042. doi:10.1007/s004540010071. MR 1797293. S2CID 14849112.
  6. ^ Page Module:Citation/CS1/styles.css has no content.Christensen, Richard M. (2013). The Theory of Materials Failure. Oxford University Press. p. 201. ISBN 978-0-19-966211-1.
  7. ^ Page Module:Citation/CS1/styles.css has no content.Ball, Philip (2016). Patterns in Nature: Why the Natural World Looks the Way It Does. University of Chicago Press. p. 15. ISBN 978-0-226-33242-0.
  8. ^ Tilings and patterns, Sec. 9.3 Other Monohedral tilings by convex polygons
  9. ^ Tilings and patterns, from list of 107 isohedral tilings, pp. 473–481
  10. ^ Tilings and patterns, uniform tilings that are not edge-to-edge
  11. ^ Order in Space: A design source book, Keith Critchlow, pp. 74–75, pattern 2
  12. ^ Coxeter, Regular Complex Polytopes, pp. 111–112, p. 136.
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Space Family A~n−1 C~n−1 B~n−1 D~n−1 G~2 / F~4 / 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
En−1 Uniform (n−1)-honeycomb 0[n] δn hδn qδn 1k2 • 2k1 • k21

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