Trihexagonal tiling

Last updated October 28, 2024

Trihexagonal tiling

Type	Semiregular tiling
Vertex configuration	(3.6)²
Schläfli symbol	r{6,3} or ${\begin{Bmatrix}6\\3\end{Bmatrix}}$ h₂{6,3}
Wythoff symbol	2 \| 6 3 3 3 \| 3
Coxeter diagram	=
Symmetry	p6m, [6,3], (*632)
Rotation symmetry	p6, [6,3]⁺, (632) p3, [3^[3]]⁺, (333)
Bowers acronym	That
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]

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

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.

The kagome pattern is common in bamboo weaving in East Asia. In 2022, archaeologists found bamboo weaving remains at the Dongsunba ruins in Chongqing, China, 200 BC. After 2200 years, the kagome pattern is still clear.^[5]^[6]

Inuit snowshoe Tagluk
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.^[7] 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.^[8] 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.^[8] 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.^[9] Quantum magnets realized on Kagome metals have been discovered to exhibit many unexpected electronic and magnetic phenomena.^[10]^[11]^[12]^[13] It is also proposed that SYK behavior can be observed in two dimensional kagome lattice with impurities.^[14]

The term is much in use nowadays in the scientific literature, especially by theorists studying the magnetic properties of a theoretical kagome lattice.

Symmetry

The trihexagonal tiling has Schläfli symbol of r{6,3}, or Coxeter diagram, , symbolizing the fact that it is a rectified hexagonal tiling, {6,3}. Its symmetries can be described by the wallpaper group p6mm, (*632),^[15] 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
fundamental domain
Wythoff	2 \| 6 3	3 3 \| 3
Coxeter		=
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.^[16] Every circle is in contact with 4 other circles in the packing (kissing number).

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)

Related quasiregular tilings

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.^[17]^[18]

*n32 orbifold symmetries of quasiregular tilings: (3.n)²
Construction	Spherical			Euclidean	Hyperbolic
Construction	*332	*432	*532	*632	*732	*832...	*∞32
Quasiregular figures
Vertex	(3.3)²	(3.4)²	(3.5)²	(3.6)²	(3.7)²	(3.8)²	(3.∞)²

Related regular complex apeirogons

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.^[19]

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

3{12}2 or	6{6}2 or

Related Research Articles

In geometry, an octahedron is a polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Regular octahedra occur in nature as crystal structures. Many types of irregular octahedra also exist, including both convex and non-convex shapes.

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

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 $t {3,6}$ .

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. Conway called it a quadrille.

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 ${3,6}.$

In geometry, the truncated trihexagonal tiling is one of eight semiregular tilings of the Euclidean plane. There are one square, one hexagon, and one dodecagon on each vertex. It has Schläfli symbol of tr{3,6}.

In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.

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 called this honeycomb a cubille.

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.

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.

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.

In geometry, the order-3 snub heptagonal tiling is a semiregular tiling of the hyperbolic plane. There are four triangles and 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.

<span class="mw-page-title-main">Snub trioctagonal tiling</span>

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}.

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.

In the field of hyperbolic geometry, the order-4 hexagonal tiling honeycomb arises as 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 flat plane in hyperbolic space that approaches a single ideal point at infinity.

In the field of hyperbolic geometry, the order-6 hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells with an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere: a flat plane in hyperbolic space that approaches a single ideal point at infinity.

The triangular tiling honeycomb is one of 11 paracompact regular space-filling tessellations in hyperbolic 3-space. It is called paracompact because it has infinite cells and vertex figures, with all vertices as ideal points at infinity. It has Schläfli symbol {3,6,3}, being composed of triangular tiling cells. Each edge of the honeycomb is surrounded by three cells, and each vertex is ideal with infinitely many cells meeting there. Its vertex figure is a hexagonal tiling.

References

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