Trihexagonal tiling

In geometry, the trihexagonal tiling is one of 11 uniform tilings of the Euclidean plane by regular polygons. 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. This pattern, and its place in the classification of uniform tilings, was already known to Johannes Kepler in his 1619 book Harmonices Mundi. 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).

Kagome lattice

The term kagome lattice was coined by the Japanese physicist Kôdi Husimi, 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, filling space by regular tetrahedra and truncated tetrahedra, has been called a hyper-kagome lattice. 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. The trihexagonal prismatic honeycomb represents its edges and vertices. Some minerals, namely jarosites and herbertsmithite, contain two-dimensional layers or three-dimensional kagome lattice arrangements 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 Co 3 V 2 O 8 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.

Models on the kagome lattice

The physical interest of the kagome lattice stems from its corner-sharing triangles: three antiferromagnetically coupled spins on a triangle cannot all be pairwise antiparallel, and since the triangles share only corners, this geometrical frustration does not propagate into a rigid order of the whole lattice but leaves a macroscopically large number of degenerate low-energy configurations. Shōji solved the Ising model on the kagome lattice exactly in 1951 by relating it to the honeycomb lattice through a star–triangle transformation; the ferromagnetic case has a phase transition at k_\mathrm{B} T_c/J = 4/\ln(3+2\sqrt{3}) \approx 2.143 . The antiferromagnetic Ising model, by contrast, does not order at any temperature and retains a finite residual entropy down to zero temperature. The spin-½ Heisenberg antiferromagnet on the kagome lattice is one of the most studied models of frustrated magnetism and a leading candidate for a quantum spin liquid ground state without magnetic order. Density matrix renormalization group calculations point to a gapped spin liquid, and inelastic neutron scattering on herbertsmithite shows a continuum of fractionalized excitations instead of the sharp spin waves of an ordered magnet. For electrons hopping between nearest neighbours, the band structure of the kagome lattice contains, besides two dispersive bands touching at Dirac points, a completely flat band: destructive interference around the hexagons allows localized states whose energy is independent of momentum, so that interactions dominate over kinetic energy. This makes the lattice a model system for flat-band ferromagnetism and correlated electron phases. In the doped Hubbard model with strong repulsion, the frustration of electron motion produces self-trapped polarons of resonating valence bonds.

Symmetry

s of p6m (*632) 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), 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.

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

Names in other languages