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 trihexagonal tiling is one of 11

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 plane. Uniform tilings are related to the f ...

s 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 of ...

by regular polygons. See in particular Theorem 2.1.3, p. 59 (classification of uniform tilings); Figure 2.1.5, p.63 (illustration of this tiling), Theorem 2.9.1, p. 103 (classification of colored tilings), Figure 2.9.2, p. 105 (illustration of colored tilings), Figure 2.5.3(d), p. 83 (topologically equivalent star tiling), and Exercise 4.1.3, p. 171 (topological equivalence of trihexagonal and two-triangle tilings). It consists of

equilateral triangle In geometry, an equilateral triangle is a triangle in which all three sides have the same length. In the familiar Euclidean geometry, an equilateral triangle is also equiangular; that is, all three internal angles are also congruent to each othe ...

s and

regular 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, arranged so that each hexagon is surrounded by triangles and vice versa. The name derives from the fact that it combines a regular

hexagonal tiling 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 or (as a truncated triangular tiling). English mathemat ...

and a regular

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 ...

. Two hexagons and two triangles alternate around each

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

, and its edges form an infinite

arrangement of lines In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orchestr ...

. Its dual is the

rhombille 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 a ...

. This pattern, and its place in the classification of uniform tilings, was already known to

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

in his 1619 book ''

Harmonices Mundi ''Harmonice Mundi (Harmonices mundi libri V)''The full title is ''Ioannis Keppleri Harmonices mundi libri V'' (''The Five Books of Johannes Kepler's The Harmony of the World''). (Latin: ''The Harmony of the World'', 1619) is a book by Johannes ...

''. The pattern has long been used in Japanese

basketry Basket weaving (also basketry or basket making) is the process of weaving or sewing pliable materials into three-dimensional artifacts, such as baskets, mats, mesh bags or even furniture. Craftspeople and artists specialized in making baskets ...

, where it is called kagome. The Japanese term for this pattern has been taken up in physics, where it is called a Kagome lattice. It occurs also in the crystal structures of certain minerals. Conway calls it a hexadeltille, combining alternate elements from a

(hextille) and

(deltille).

Kagome

Kagome ( ja, 籠目) is a traditional Japanese woven bamboo pattern; its name is composed from the words ''kago'', meaning "basket", and ''me'', meaning "eye(s)", referring to the pattern of holes in a woven basket. Kagome lattice blue

It is a

woven Woven fabric is any textile formed by weaving. Woven fabrics are often created on a loom, and made of many threads woven on a warp and a weft. Technically, a woven fabric is any fabric made by interlacing two or more threads at right angles to on ...

arrangement In music, an arrangement is a musical adaptation of an existing composition. Differences from the original composition may include reharmonization, melodic paraphrasing, orchestration, or formal development. Arranging differs from orches ...

lath A lath or slat is a thin, narrow strip of straight-grained wood used under roof shingles or tiles, on lath and plaster walls and ceilings to hold plaster, and in lattice and trellis work. ''Lath'' has expanded to mean any type of backing mate ...

s composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The

process gives the Kagome a chiral

wallpaper group A wallpaper is a mathematical object covering a whole Euclidean plane by repeating a motif indefinitely, in manner that certain isometries keep the drawing unchanged. To a given wallpaper there corresponds a group of such congruent transformatio ...

symmetry, p6, (632).

Kagome lattice

The term kagome lattice was coined by Japanese physicist

Kôdi Husimi Kōji Husimi (June 29, 1909 – May 8, 2008, ja, 伏見康治) was a Japanese theoretical physicist who served as the president of the Science Council of Japan.. Husimi trees in graph theory, the Husimi Q representation in quantum mechanics, an ...

, and first appeared in a 1951 paper by his assistant Ichirō Shōji. The kagome lattice in this sense consists of the vertices and edges of the trihexagonal tiling. Despite the name, these crossing points do not form a mathematical lattice. A related three dimensional structure formed by the vertices and edges of the

quarter cubic honeycomb The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is c ...

, filling space by regular

tetrahedra In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the o ...

and

truncated tetrahedra In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular tetrahedron ...

, has been called a ''hyper-kagome lattice''. It is represented by the vertices and edges of the

, filling space by regular

and

. It contains four sets of parallel planes of points and lines, each plane being a two dimensional kagome lattice. A second expression in three dimensions has parallel layers of two dimensional lattices and is called an ''orthorhombic-kagome lattice''. The

trihexagonal prismatic honeycomb The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into pri ...

represents its edges and vertices. Some

mineral In geology and mineralogy, a mineral or mineral species is, broadly speaking, a solid chemical compound with a fairly well-defined chemical composition and a specific crystal structure that occurs naturally in pure form.John P. Rafferty, ed. ( ...

s, namely

jarosite Jarosite is a basic hydrous sulfate of potassium and ferric iron (Fe-III) with a chemical formula of KFe3(SO4)2(OH)6. This sulfate mineral is formed in ore deposits by the oxidation of iron sulfides. Jarosite is often produced as a byproduct duri ...

s and

herbertsmithite Herbertsmithite is a mineral with chemical structure Zn Cu3( OH)6 Cl2. It is named after the mineralogist Herbert Smith (1872–1953) and was first found in 1972 in Chile. It is polymorphous with kapellasite and closely related to paratacamit ...

, contain two-dimensional layers or three-dimensional kagome lattice arrangement of

atom Every atom is composed of a nucleus and one or more electrons bound to the nucleus. The nucleus is made of one or more protons and a number of neutrons. Only the most common variety of hydrogen has no neutrons. Every solid, liquid, gas, and ...

s in their

crystal structure In crystallography, crystal structure is a description of the ordered arrangement of atoms, ions or molecules in a crystal, crystalline material. Ordered structures occur from the intrinsic nature of the constituent particles to form symmetric pat ...

. These minerals display novel physical properties connected with

geometrically frustrated magnet In condensed matter physics, the term geometrical frustration (or in short: frustration) refers to a phenomenon where atoms tend to stick to non-trivial positions or where, on a regular crystal lattice, conflicting inter-atomic forces (each one favo ...

ism. For instance, the spin arrangement of the magnetic ions in Co₃V₂O₈ rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures. Quantum magnets realized on Kagome metals have been discovered to exhibit many unexpected electronic and magnetic phenomena. It is also proposed that SYK behavior can be observed in two dimensional kagome lattice with impurities. The term is much in use nowadays in the scientific literature, especially by theorists studying the magnetic properties of a theoretical kagome lattice. See also:

Kagome crest The Kagome crest or is a star-shaped emblem related to the kagome lattice design ( Hexagonal and Octagonal lattices). The Kagome mon can be depicted as, either, a six-pointed star (a hexagram) and as an eight-pointed star (an octagram): # ...

Symmetry

Tiling Dual Semiregular V4-6-12 Bisected Hexagonal

The trihexagonal tiling 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 more ...

of r, or

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 to ...

, , symbolizing the fact that it is a rectified

, . Its

symmetries Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

can be described by the

p6mm, (*632), and the tiling can be derived as a

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 ...

within the reflectional

fundamental domain Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...

s of this group. The trihexagonal tiling is a

quasiregular tiling In geometry, a quasiregular polyhedron is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the sem ...

, alternating two types of polygons, with

vertex configuration In geometry, a vertex configurationCrystallography ...

(3.6)². It is also a

, one of eight derived from the regular hexagonal tiling.

Uniform colorings

There are two distinct

uniform coloring In geometry, a uniform coloring is a property of a uniform figure (uniform tiling or uniform polyhedron) that is colored to be vertex-transitive. Different symmetries can be expressed on the same geometric figure with the faces following differ ...

s of a trihexagonal tiling. Naming the colors by indices on the 4 faces around a vertex (3.6.3.6): 1212, 1232. The second is called a cantic

, h₂, with two colors of triangles, existing in p3m1 (*333) symmetry.

Circle packing

The trihexagonal tiling can be used as a

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 ...

, placing equal diameter circles at the center of every point. Every circle is in contact with 4 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 of ...

). :

Topologically equivalent tilings

The ''trihexagonal tiling'' can be geometrically distorted into topologically equivalent tilings of lower symmetry. In these variants of the tiling, the edges do not necessarily line up to form straight lines.

Related quasiregular tilings

The ''trihexagonal tiling'' exists in a sequence of symmetries of quasiregular tilings with

vertex configuration In geometry, a vertex configurationCrystallography ...

s (3.''n'')², progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With

orbifold notation In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curvature. The advanta ...

symmetry of *''n''32 all of these tilings are

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 ...

within a

of symmetry, with generator points at the right angle corner of the domain.

Related regular complex apeirogons

There are 2

regular complex apeirogon In geometry, a complex polytope is a generalization of a polytope in real space to an analogous structure in a complex Hilbert space, where each real dimension is accompanied by an imaginary one. A complex polytope may be understood as a collect ...

s, sharing the vertices of the trihexagonal 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 arranged like a

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

, and

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 (geometry), vertex of a polyhedron. Mark a point somewhere along each connect ...

s are ''r''-gonal. The first is made of triangular edges, two around every vertex, second has hexagonal edges, two around every vertex.