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Hexagonal tiling

From Wikipedia, the free encyclopedia

Hexagonal tiling
Hexagonal tiling

Type Regular tiling
Vertex configuration 6.6.6 (or 63)
Tiling 6 vertfig.svg
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) CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png
CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png
CDel node 1.pngCDel split1.pngCDel branch 11.png
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 three[clarification needed] 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

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
Uniform tiling 63-t0.svg
Uniform tiling 63-t12.svg
Uniform tiling 333-t012.svg
Truncated rhombille tiling.png
Hexagonal tiling 4-colors.svg
Hexagonal tiling 2-1.svg
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 CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 6.pngCDel node.png CDel node 1.pngCDel split1.pngCDel branch 11.png
Conway H 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)
Uniform tiling 63-t0.svg
Chamfered hexagonal tiling.png
Truncated rhombille tiling.png
Chamfered hexagonal tiling2.png
Rhombic star tiling.png

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
1-uniform n1.svg

Original
Regular hexagon.svg

Vertex type 3-3-3-3-3-3.svg
2-uniform n10.svg

1/3 dissected
2-uniform n19.svg

2/3 dissected
1-uniform n11.svg

fully dissected
Regular Tiling Inset 2-Uniform Duals Regular Tiling
Dual of Planar Tiling (Uniform Regular 2) 6.6.6.png

Original
Insetting Polygon for Uniform Tilings 1.png
Dual of Planar Tiling (Uniform Two 8) 3.3.3.3.3.3; 3.3.6.6.png

1/3 inset
Dual of Planar Tiling (Uniform Two 9) 36; 34.6 1.png

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.

Kah 3 6 romb.png

Rhombic tiling
Uniform tiling 63-t0.svg

Hexagonal tiling
Chicken Wire close-up.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:

Pent-Hex-Type1-2.png

Pentagonal tiling type 1 with overlays of regular hexagons (each comprising 2 pentagons).
Pent-Hex-Type3-3.png

pentagonal tiling type 3 with overlays of regular hexagons (each comprising 3 pentagons).
Pent-Hex-Type4-4.png

Pentagonal tiling type 4 with overlays of semiregular hexagons (each comprising 4 pentagons).
Pent-Hex-Type3-9.png

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 CDel node 1.pngCDel 6.pngCDel node.pngCDel n.pngCDel node.png, progressing to infinity.

*n62 symmetry mutation of regular tilings: {6,n}
Spherical Euclidean Hyperbolic tilings
Hexagonal dihedron.svg

{6,2}
Uniform tiling 63-t0.svg

{6,3}
H2 tiling 246-1.png

{6,4}
H2 tiling 256-1.png

{6,5}
H2 tiling 266-4.png

{6,6}
H2 tiling 267-4.png

{6,7}
H2 tiling 268-4.png

{6,8}
...
H2 tiling 26i-4.png

{6,∞}

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

*n32 symmetry mutation of regular tilings: {n,3}
Spherical Euclidean Compact hyperb. Paraco. Noncompact hyperbolic
Spherical trigonal hosohedron.png
Uniform tiling 332-t0.png
Uniform tiling 432-t0.png
Uniform tiling 532-t0.png
Uniform polyhedron-63-t0.png
Heptagonal tiling.svg
H2-8-3-dual.svg
H2-I-3-dual.svg
H2 tiling 23j12-1.png
H2 tiling 23j9-1.png
H2 tiling 23j6-1.png
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.

*n32 symmetry mutation of truncated tilings: n.6.6
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
Hexagonal dihedron.svg
Uniform tiling 332-t12.png
Uniform tiling 432-t12.png
Uniform tiling 532-t12.png
Uniform tiling 63-t12.svg
Truncated order-7 triangular tiling.svg
H2-8-3-trunc-primal.svg
H2 tiling 23i-6.png
H2 tiling 23j12-6.png
H2 tiling 23j9-6.png
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
Hexagonal Hosohedron.svg
Spherical triakis tetrahedron.png
Spherical tetrakis hexahedron.png
Spherical pentakis dodecahedron.png
Uniform tiling 63-t2.svg
Heptakis heptagonal tiling.svg
H2-8-3-kis-dual.svg
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 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.

Symmetry mutations of dual quasiregular tilings: V(3.n)2
*n32 Spherical Euclidean Hyperbolic
*332 *432 *532 *632 *732 *832... *∞32
Tiling
Uniform tiling 432-t0.png
Spherical rhombic dodecahedron.png
Spherical rhombic triacontahedron.png
Rhombic star tiling.png
7-3 rhombille tiling.svg
H2-8-3-rhombic.svg
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

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.)

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}
CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg
Uniform tiling 63-t0.svg
Uniform tiling 63-t01.svg
Uniform tiling 63-t1.svg
Uniform tiling 63-t12.svg
Uniform tiling 63-t2.svg
Uniform tiling 63-t02.png
Uniform tiling 63-t012.svg
Uniform tiling 63-snub.png
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
P6-type1.png
P6-type2.png
P6-type2-chiral coloring.png
P6-type3.png
Prototile p6-type1.png

b=e
B+C+D=360°
Prototile p6-type2.png

b=e, d=f
B+C+E=360°
Prototile p6-type3.png

a=f, b=c, d=e
B=D=F=120°
Lattice p6-type1.png

2-tile lattice
Lattice p6-type2.png

4-tile lattice
Lattice p6-type3.png

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*)
Isohedral tiling p6-1.png
Isohedral tiling p6-2.png
Isohedral tiling p6-3.png
Isohedral tiling p6-6.png
Isohedral tiling p6-9.png
Isohedral tiling p6-10.png
pgg (22×) p31m (3*3) p2 (2222) cmm (2*22) p6m (*632)
Isohedral tiling p6-4.png
Isohedral tiling p6-5.png
Isohedral tiling p6-8.png
Isohedral tiling p6-11.png
Isohedral tiling p6-7.png
Isohedral tiling p6-12.png
Isohedral tiling p6-13.png

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)
Isohedral tiling p4-18.png

Parallelogram
Isohedral tiling p4-20.png

Trapezoid
Isohedral tiling p4-19.png

Parallelogram
Isohedral tiling p4-19b.png

Rectangle
Isohedral tiling p4-17.png

Parallelogram
Isohedral tiling p4-21.png

Rectangle
Isohedral tiling p4-22.png

Rectangle
Isohedrally-tiled pentagons
p2 (2222) pgg (22×) p3 (333)
P5-type1.png
P5-type2.png
P5-type3.png

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

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.

1-uniform-1-circlepack.svg

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, 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.

Complex apeirogon 2-12-3.png
Complex apeirogon 6-4-3.png
Truncated complex polygon 6-6-2.png
2{12}3 or CDel node 1.pngCDel 12.pngCDel 3node.png 6{4}3 or CDel 6node 1.pngCDel 4.pngCDel 3node.png CDel 6node 1.pngCDel 6.pngCDel node 1.png

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 [1]

External links

Space Family / /
E2 Uniform tiling {3[3]} δ3 3 3 Hexagonal
E3 Uniform convex honeycomb {3[4]} δ4 4 4
E4 Uniform 4-honeycomb {3[5]} δ5 5 5 24-cell honeycomb
E5 Uniform 5-honeycomb {3[6]} δ6 6 6
E6 Uniform 6-honeycomb {3[7]} δ7 7 7 222
E7 Uniform 7-honeycomb {3[8]} δ8 8 8 133331
E8 Uniform 8-honeycomb {3[9]} δ9 9 9 152251521
E9 Uniform 9-honeycomb {3[10]} δ10 10 10
En-1 Uniform (n-1)-honeycomb {3[n]} δn n n 1k22k1k21
This page was last edited on 21 December 2020, at 20:00
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