geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the hexagonal tiling or hexagonal tessellation is a

regular tiling Euclidean Plane (mathematics), plane Tessellation, tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Johannes Kepler, Kepler in his ''Harmonices Mundi'' (Latin langua ...

of the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...

, in which exactly three

hexagon In geometry, a hexagon (from Greek , , meaning "six", and , , meaning "corner, angle") is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°. Regular hexagon A '' regular hexagon'' has ...

s meet at each vertex. It has

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to mo ...

of or (as a truncated triangular tiling). English mathematician John Conway called it a hextille. The

internal angle In geometry, an angle of a polygon is formed by two sides of the polygon that share an endpoint. For a simple (non-self-intersecting) polygon, regardless of whether it is convex or non-convex, this angle is called an interior angle (or ) if ...

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 In geometry, the triangular tiling or triangular tessellation is one of the three regular tilings of the Euclidean plane, and is the only such tiling where the constituent shapes are not parallelogons. Because the internal angle of the equilater ...

and the

square tiling In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of meaning it has 4 squares around every vertex. Conway called it a quadrille. The internal angle of th ...

Applications

The hexagonal tiling is the densest way to arrange circles in two dimensions. The

honeycomb conjecture The honeycomb conjecture states that a regular hexagonal grid or honeycomb has the least total perimeter of any subdivision of the plane into regions of equal area. The conjecture was proven in 1999 by mathematician Thomas C. Hales. Theorem Le ...

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 William Thomson, 1st Baron Kelvin, (26 June 182417 December 1907) was a British mathematician, mathematical physicist and engineer born in Belfast. Professor of Natural Philosophy at the University of Glasgow for 53 years, he did important ...

, who believed that the

Kelvin structure The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and p ...

(or

body-centered cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of ...

lattice) is optimal. However, the less regular

Weaire–Phelan structure In geometry, the Weaire–Phelan structure is a three-dimensional structure representing an idealised foam of equal-sized bubbles, with two different shapes. In 1993, Denis Weaire and Robert Phelan found that this structure was a better solution ...

is slightly better. This structure exists naturally in the form of

graphite Graphite () is a crystalline form of the element carbon. It consists of stacked layers of graphene. Graphite occurs naturally and is the most stable form of carbon under standard conditions. Synthetic and natural graphite are consumed on la ...

, where each sheet of

graphene Graphene () is an allotrope of carbon consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice nanostructure.

resembles chicken wire, with strong covalent carbon bonds. Tubular graphene sheets have been synthesised; these are known as

carbon nanotube A scanning tunneling microscopy image of a single-walled carbon nanotube Rotating single-walled zigzag carbon nanotube A carbon nanotube (CNT) is a tube made of carbon with diameters typically measured in nanometers. ''Single-wall carbon na ...

s. They have many potential applications, due to their high

tensile strength Ultimate tensile strength (UTS), often shortened to tensile strength (TS), ultimate strength, or F_\text within equations, is the maximum stress that a material can withstand while being stretched or pulled before breaking. In brittle materials ...

and electrical properties. Silicene is similar.

Chicken wire Chicken wire, or poultry netting, is a mesh of wire commonly used to fence in fowl, such as chickens, in a run or coop. It is made of thin, flexible, galvanized steel wire with hexagonal gaps. Available in 1 inch (about 2.5 cm) diameter, ...

consists of a hexagonal lattice (often not regular) of wires. File:Kissing-2d.svg, The densest

circle packing In geometry, circle packing is the study of the arrangement of circles (of equal or varying sizes) on a given surface such that no overlapping occurs and so that no circle can be enlarged without creating an overlap. The associated '' packing de ...

is arranged like the hexagons in this tiling File:Chicken Wire close-up.jpg,

fencing File:Graphene xyz.jpg,

Graphene Graphene () is an allotrope of carbon consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice nanostructure.

File:Carbon nanotube zigzag povray.PNG, A

can be seen as a hexagon tiling on a Cylinder (geometry), cylindrical surface File:Tile (AM 1955.117-1).jpg, alt=Hexagonal tile with blue bird and flowers, Hexagonal Persian tile c.1955 The hexagonal tiling appears in many crystals. In three dimensions, the

face-centered cubic In crystallography, the cubic (or isometric) crystal system is a crystal system where the unit cell is in the shape of a cube. This is one of the most common and simplest shapes found in crystals and minerals. There are three main varieties of ...

and

hexagonal close packing In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occu ...

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 Copper is a chemical element with the symbol Cu (from la, cuprum) and atomic number 29. It is a soft, malleable, and ductile metal with very high thermal and electrical conductivity. A freshly exposed surface of pure copper has a pink ...

, 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 construction In geometry, a Wythoff construction, named after mathematician Willem Abraham Wythoff, is a method for constructing a uniform polyhedron or plane tiling. It is often referred to as Wythoff's kaleidoscopic construction. Construction process ...

s. 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 _''h'',''k'', and can be applied to hyperbolic tilings for ''p'' > 6. The 3-color tiling is a tessellation generated by the order-3 permutohedrons.

Chamfered hexagonal tiling

chamfered A chamfer or is a transitional edge between two faces of an object. Sometimes defined as a form of bevel, it is often created at a 45° angle between two adjoining right-angled faces. Chamfers are frequently used in machining, carpentry, fur ...

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 In geometry, the rhombille tiling, also known as tumbling blocks, reversible cubes, or the dice lattice, is a tessellation of identical 60° rhombi on the Euclidean plane. Each rhombus has two 60° and two 120° angles; rhombi with this shape ar ...

Related tilings

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

2-uniform tiling A ''k''-uniform tiling is a tiling of Tessellation, tilings of the plane by convex regular polygons, connected edge-to-edge, with ''k'' types of vertices. The 1-uniform tiling include 3 regular tilings, and 8 semiregular tilings. A 1-uniform tilin ...

s, and the

: 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 In geometry, the rhombic dodecahedron is a convex polyhedron with 12 congruent rhombic faces. It has 24 edges, and 14 vertices of 2 types. It is a Catalan solid, and the dual polyhedron of the cuboctahedron. Properties The rhombic dodecahed ...

and the

rhombo-hexagonal dodecahedron In geometry, the elongated dodecahedron, extended rhombic dodecahedron, rhombo-hexagonal dodecahedron or hexarhombic dodecahedron is a convex dodecahedron with 8 rhombic and 4 hexagonal faces. The hexagons can be made equilateral, or regular de ...

tessellations in 3 dimensions. It is also possible to subdivide the prototiles of certain hexagonal tilings by two, three, four or nine equal pentagons:

Symmetry mutations

This tiling is topologically related as a part of sequence of regular tilings with

al faces, starting with the hexagonal tiling, with

, and

Coxeter diagram Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

, progressing to infinity. This tiling is topologically related to regular polyhedra with

vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...

''n''³, as a part of sequence that continues into the

hyperbolic plane In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ' ...

. 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 ,3

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...

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 In geometry, a uniform polyhedron has regular polygons as faces and is vertex-transitive (i.e., there is an isometry mapping any vertex onto any other). It follows that all vertices are congruent. Uniform polyhedra may be regular (if also ...

there are eight

uniform tiling In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive. Uniform tilings can exist in both the Euclidean plane and Hyperbolic space, hyperbolic plane. Uniform tilings ar ...

s that can be based from the regular hexagonal tiling (or the dual

). 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. They are all isohedral. Each has parametric variations within a fixed symmetry. Type 2 contains

glide reflection In 2-dimensional geometry, a glide reflection (or transflection) is a symmetry operation that consists of a reflection over a line and then translation along that line, combined into a single operation. The intermediate step between reflecti ...

s, and is 2-isohedral keeping chiral pairs distinct.

Topologically equivalent tilings

Hexagonal tilings can be made with the identical 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. Single-color (1-tile) lattices are parallelogon hexagons. Other isohedrally-tiled topological hexagonal tilings are seen as quadrilaterals and pentagons that are not edge-to-edge, but interpreted as colinear adjacent edges: 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. It can also be distorted into a

chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...

4-colored tri-directional weaved pattern, distorting some hexagons into

parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of eq ...

s. The weaved pattern with 2 colored faces have rotational 632 (p6) symmetry. A

chevron Chevron (often relating to V-shaped patterns) may refer to: Science and technology * Chevron (aerospace), sawtooth patterns on some jet engines * Chevron (anatomy), a bone * '' Eulithis testata'', a moth * Chevron (geology), a fold in rock la ...

pattern has pmg (22*) symmetry, which is lowered to p1 (°) with 3 or 4 colored tiles.

Circle packing

The hexagonal tiling can be used as a

, placing equal diameter circles at the center of every point. Every circle is in contact with 3 other circles in the packing (

kissing number In geometry, the kissing number of a mathematical space is defined as the greatest number of non-overlapping unit spheres that can be arranged in that space such that they each touch a common unit sphere. For a given sphere packing (arrangement o ...

).Order in Space: A design source book, Keith Critchlow, pp. 74–75, pattern 2 The gap inside each hexagon allows for one circle, creating the densest packing from the

, with each circle contact with the maximum of 6 circles. : 1-uniform-1-circlepack

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'r'' are constrained by: 1/''p'' + 2/''q'' + 1/''r'' = 1. Edges have ''p'' vertices, and vertex figures are ''r''-gonal.Coxeter, Regular Complex Polytopes, pp. 111–112, p. 136. 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.

References

* Coxeter, H.S.M. ''

Regular Polytopes In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

'', (3rd edition, 1973), Dover edition, p. 296, Table II: Regular honeycombs * (Chapter 2.1: ''Regular and uniform tilings'', pp. 58–65) * * John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, ''The Symmetries of Things'' 2008,

External links

* ** ** * {{Tessellation Euclidean tilings Isogonal tilings Isohedral tilings Regular tilings Regular tessellations