Hexagonal tiling

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Short description: Regular tiling of the plane

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.


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 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 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 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)
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 Inset Dual Tilings
1-uniform n1.svg
Regular hexagon.svgVertex 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
Inset Variations of Dual Uniform Tiling.svg 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.

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:

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 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.

This tiling is topologically related to regular polyhedra with vertex figure n3, as a part of 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 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.

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.)

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 120px 120px 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
Isohedral tiling p4-20.png
Isohedral tiling p4-19.png
Isohedral tiling p4-19b.png
Isohedral tiling p4-17.png
Isohedral tiling p4-21.png
Isohedral tiling p4-22.png
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.


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


  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.

External links

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 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