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In geometry, the trihexagonal tiling is one of 11 of the Euclidean plane 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 triangles and , 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 tiling 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). (2025). 9781568812205, A K Peters, Ltd.. ISBN 9781568812205
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.
It is a Weaving arrangement of composed of interlaced triangles such that each point where two laths cross has four neighboring points, forming the pattern of a trihexagonal tiling. The Weaving process gives the Kagome a chiral wallpaper group symmetry, p6 (632).
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 , namely and herbertsmithite, contain two-dimensional layers or three-dimensional kagome lattice arrangement of 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. Quantum magnets realized on Kagome metal 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.
The first is made of triangular edges, two around every vertex, second has hexagonal edges, two around every vertex.
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