Trihexagonal tiling

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Trihexagonal tiling
Tiling 3-6 simple.svg
Type Semiregular tiling
Vertex configuration Tiling 3-6 vertfig.svg
(3.6)2
Schläfli symbol r{6,3} or
h2{6,3}
Wythoff symbol 6 3
3 3 | 3
Coxeter diagram CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png
CDel branch 10ru.pngCDel split2.pngCDel node 1.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry p6m, [6,3], (*632)
Rotation symmetry p6, [6,3]+, (632)
p3, [3[3]]+, (333)
Bowers acronymThat
Dual Rhombille tiling
Properties Vertex-transitive Edge-transitive

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. [1] It consists of equilateral triangles and regular hexagons, 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 and a regular triangular tiling. Two hexagons and two triangles alternate around each vertex, and its edges form an infinite arrangement of lines. Its dual is the rhombille tiling. [2]

Contents

This pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi . [3] The pattern has long been used in Japanese basketry, 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 hexagonal tiling (hextille) and triangular tiling (deltille). [4]

Kagome

Japanese basket showing the kagome pattern P12fig1.jpg
Japanese basket showing the kagome pattern

Kagome (Japanese : 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 pattern in detail Kagome lattice blue.svg
kagome pattern in detail

It is a woven arrangement of laths composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The woven process gives the Kagome a chiral wallpaper group symmetry, p6, (632).

Kagome lattice

The term kagome lattice was coined by Japanese physicist Kôdi Husimi, and first appeared in a 1951 paper by his assistant Ichirō Shōji. [5] 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, filling space by regular tetrahedra and truncated tetrahedra, has been called a hyper-kagome lattice. [6] It is represented by the vertices and edges of the quarter cubic honeycomb, filling space by regular tetrahedra and truncated tetrahedra. 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. [6] The trihexagonal prismatic honeycomb represents its edges and vertices.

Some minerals, namely jarosites and herbertsmithite, contain two-dimensional layers or three-dimensional kagome lattice arrangement of atoms in their crystal structure. These minerals display novel physical properties connected with geometrically frustrated magnetism. For instance, the spin arrangement of the magnetic ions in Co3V2O8 rests in a kagome lattice which exhibits fascinating magnetic behavior at low temperatures. [7] Quantum magnets realized on Kagome metals have been discovered to exhibit many unexpected electronic and magnetic phenomena. [8] [9] [10] [11] It is also proposed that SYK behavior can be observed in two dimensional kagome lattice with impurities. [12]

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 crests.

Symmetry

30-60-90 triangle fundamental domains of p6m (*632) symmetry Tiling Dual Semiregular V4-6-12 Bisected Hexagonal.svg
30-60-90 triangle fundamental domains of p6m (*632) symmetry

The trihexagonal tiling has Schläfli symbol of r{6,3}, or Coxeter diagram, CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.png, symbolizing the fact that it is a rectified hexagonal tiling, {6,3}. Its symmetries can be described by the wallpaper group p6mm, (*632), [13] and the tiling can be derived as a Wythoff construction within the reflectional fundamental domains of this group. The trihexagonal tiling is a quasiregular tiling, alternating two types of polygons, with vertex configuration (3.6)2. It is also a uniform tiling, one of eight derived from the regular hexagonal tiling.

Uniform colorings

There are two distinct uniform colorings of a trihexagonal tiling. Naming the colors by indices on the 4 faces around a vertex (3.6.3.6): 1212, 1232. [1] The second is called a cantic hexagonal tiling , h2{6,3}, with two colors of triangles, existing in p3m1 (*333) symmetry.

Symmetry p6m, (*632)p3m, (*333)
Coloring Uniform polyhedron-63-t1.png Uniform tiling 333-t12.png
fundamental
domain
632 fundamental domain t1.png 333 fundamental domain t01.png
Wythoff 6 33
Coxeter CDel node.pngCDel 6.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel branch 10ru.pngCDel split2.pngCDel node 1.png = CDel node h1.pngCDel 6.pngCDel node.pngCDel 3.pngCDel node 1.png
Schläfli r{6,3}r{3[3]} = h2{6,3}

Circle packing

The trihexagonal tiling can be used as a circle packing, placing equal diameter circles at the center of every point. [14] Every circle is in contact with 4 other circles in the packing (kissing number).

1-uniform-7-circlepack.svg

Topologically equivalent tilings

The trihexagonal tiling can be geometrically distorted into topologically equivalent tilings of lower symmetry. [1] In these variants of the tiling, the edges do not necessarily line up to form straight lines.

p3m1, (*333)p3, (333)p31m, (3*3)cmm, (2*22)
Trihexagonal tiling unequal.png 3-9-star-tiling.png Hex-hexstar-tiling.svg Trihexagonal tiling unequal2.svg Distorted trihexagonal tiling.png Triangle and triangular star tiling.png Trihexagonal tiling in square tiling.svg

The trihexagonal tiling exists in a sequence of symmetries of quasiregular tilings with vertex configurations (3.n)2, progressing from tilings of the sphere to the Euclidean plane and into the hyperbolic plane. With orbifold notation symmetry of *n32 all of these tilings are wythoff construction within a fundamental domain of symmetry, with generator points at the right angle corner of the domain. [15] [16]

*n32 orbifold symmetries of quasiregular tilings: (3.n)2
Quasiregular fundamental domain.png
Construction
Spherical EuclideanHyperbolic
*332*432*532*632*732*832...*32
Quasiregular
figures
Uniform tiling 332-t1-1-.png Uniform tiling 432-t1.png Uniform tiling 532-t1.png Uniform tiling 63-t1.svg Triheptagonal tiling.svg H2-8-3-rectified.svg H2 tiling 23i-2.png
Vertex (3.3)2 (3.4)2 (3.5)2 (3.6)2 (3.7)2 (3.8)2 (3.)2

There are 2 regular complex apeirogons, 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{q}r are constrained by: 1/p + 2/q + 1/r = 1. Edges have p vertices arranged like a regular polygon, and vertex figures are r-gonal. [17]

The first is made of triangular edges, two around every vertex, second has hexagonal edges, two around every vertex.

Complex apeirogon 3-12-2.png Complex apeirogon 6-6-2.png
3{12}2 or CDel 3node 1.pngCDel 12.pngCDel node.png6{6}2 or CDel 6node 1.pngCDel 6.pngCDel node.png

See also

Related Research Articles

Cuboctahedron polyhedron with 8 triangular faces and 6 square faces

A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is the only radially equilateral convex polyhedron.

Octahedron Polyhedron with 8 triangular faces

In geometry, an octahedron is a polyhedron with eight faces, twelve edges, and six vertices. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex.

Runcinated 5-cell

In four-dimensional geometry, a runcinated 5-cell is a convex uniform 4-polytope, being a runcination of the regular 5-cell.

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 {6,3} or t{3,6}.

Square tiling Regular tiling of the Euclidean plane

In geometry, the square tiling, square tessellation or square grid is a regular tiling of the Euclidean plane. It has Schläfli symbol of {4,4}, meaning it has 4 squares around every vertex.

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 equilateral triangle is 60 degrees, six triangles at a point occupy a full 360 degrees. The triangular tiling has Schläfli symbol of {3,6}.

Snub trihexagonal tiling

In geometry, the snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol of sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.

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 are sometimes also called diamonds. Sets of three rhombi meet at their 120° angles, and sets of six rhombi meet at their 60° angles.

Cubic honeycomb

The cubic honeycomb or cubic cellulation is the only proper regular space-filling tessellation in Euclidean 3-space, made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway calls this honeycomb a cubille.

Tetrahedral-octahedral honeycomb Quasiregular space-filling tesselation

The tetrahedral-octahedral honeycomb, alternated cubic honeycomb is a quasiregular space-filling tessellation in Euclidean 3-space. It is composed of alternating regular octahedra and tetrahedra in a ratio of 1:2.

Quarter cubic honeycomb

The quarter cubic honeycomb, quarter cubic cellulation or bitruncated alternated cubic honeycomb is a space-filling tessellation in Euclidean 3-space. It is composed of tetrahedra and truncated tetrahedra in a ratio of 1:1. It is called "quarter-cubic" because its symmetry unit – the minimal block from which the pattern is developed by reflections – is four times that of the cubic honeycomb.

Triangular prismatic honeycomb

The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation in Euclidean 3-space. It is composed entirely of triangular prisms.

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 semiregular, which are merely vertex-transitive.

Rhombitriheptagonal tiling

In geometry, the rhombitriheptagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one heptagon, alternating between two squares. The tiling has Schläfli symbol rr{7, 3}. It can be seen as constructed as a rectified triheptagonal tiling, r{7,3}, as well as an expanded heptagonal tiling or expanded order-7 triangular tiling.

Snub triheptagonal tiling

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one heptagon on each vertex. It has Schläfli symbol of sr{7,3}. The snub tetraheptagonal tiling is another related hyperbolic tiling with Schläfli symbol sr{7,4}.

In geometry, a uniform tiling is a tessellation of the plane by regular polygon faces with the restriction of being vertex-transitive.

Snub trioctagonal tiling

In geometry, the order-3 snub octagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles, one octagon on each vertex. It has Schläfli symbol of sr{8,3}.

Rhombitrioctagonal tiling

In geometry, the rhombitrioctagonal tiling is a semiregular tiling of the hyperbolic plane. At each vertex of the tiling there is one triangle and one octagon, alternating between two squares. The tiling has Schläfli symbol rr{8,3}. It can be seen as constructed as a rectified trioctagonal tiling, r{8,3}, as well as an expanded octagonal tiling or expanded order-8 triangular tiling.

Hexagonal tiling honeycomb

In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.

References

  1. 1 2 3 Grünbaum, Branko; Shephard, G. C. (1987). Tilings and Patterns . W. H. Freeman. ISBN   978-0-7167-1193-3. 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).
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Further reading