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| triangular lattice | |
|---|---|
| Name | Triangular lattice |
| Symmetry | Hexagonal (6-fold) |
triangular lattice
The triangular lattice is a two-dimensional periodic arrangement of points in which each point has six equidistant nearest neighbors, forming an equilateral-triangle tiling. It appears in classical works on crystallography and solid state physics and is central to studies in condensed matter, statistical mechanics, and discrete geometry. Historically influential in the development of lattice theory, the pattern features in analyses by mathematicians and physicists associated with institutions such as École Normale Supérieure, Trinity College, Cambridge, and Kavli Institute for Theoretical Physics.
The structure is generated by two noncollinear basis vectors producing a planar Bravais lattice with sixfold rotational symmetry closely related to descriptions in texts from Augustin-Jean Fresnel era optics and later formal treatments at University of Göttingen and Princeton University. Each lattice point has coordination number six and the packing corresponds to the densest circle packing in two dimensions as treated in proofs pioneered by researchers affiliated with University of Cambridge and Harvard University. Metric invariants such as nearest-neighbor distance and areal density are standard parameters used in studies at Bell Labs and Massachusetts Institute of Technology.
Geometrically, the pattern realizes the wallpaper group p6m, a symmetry class catalogued in classifications used by curators at the Victoria and Albert Museum and by mathematicians at Institut Henri Poincaré. The lattice supports point group D6 and can be mapped to the regular hexagonal tiling discussed in treatises by authors linked to Royal Society publications and analyses by researchers from Max Planck Society. Geometric operations—rotations, reflections, glide reflections—are described in the context of symmetry operations studied at CERN and in monographs from Oxford University Press.
A convenient basis comprises two vectors of equal length meeting at 60°, a representation used in computational models developed at Los Alamos National Laboratory and software from National Institute of Standards and Technology. Coordinates are often expressed in both Cartesian and oblique bases, analogous to conventions in textbooks authored by scholars connected with California Institute of Technology and University of Chicago. Representations facilitate boundary condition choices in simulations by groups at Argonne National Laboratory and lattice-sum evaluations appearing in work at Imperial College London.
The reciprocal lattice is also a triangular lattice rotated by 90° relative to the direct lattice, a fact exploited in band-structure calculations by researchers at Brookhaven National Laboratory and in angle-resolved photoemission studies at facilities such as Stanford Linear Accelerator Center. The first Brillouin zone is hexagonal; high-symmetry points commonly denoted in condensed-matter literature appear in publications from University of California, Berkeley and Columbia University. Electronic dispersion and phonon spectra on this lattice are analyzed in methods developed at IBM Research and by theorists affiliated with École Polytechnique.
Viewed as a graph, the lattice yields planar, 6-regular graphs studied in combinatorics by researchers associated with Princeton University and combinatorialists at University of Waterloo. Enumeration problems—matchings, colorings, independent sets—trace to work presented at conferences run by Association for Computing Machinery and results published via Springer Science+Business Media venues. Rigorous results on percolation thresholds and connectivity use techniques originating from collaborations involving Courant Institute and researchers from Duke University.
The lattice underpins models of magnetism such as the classical and quantum Heisenberg and Ising models analyzed in landmark studies at Los Alamos National Laboratory, Bell Labs, and University of Illinois Urbana-Champaign. It describes atomic arrangements in materials like graphene analogues and monolayer transition-metal dichalcogenides explored at National Graphene Institute and in experiments at Lawrence Berkeley National Laboratory. Studies of superfluidity, Bose–Einstein condensation in optical lattices, and cold-atom simulators involve setups used at MIT and Max Planck Institute for Quantum Optics.
Generalizations include anisotropic distortions leading to oblique or rectangular lattices treated in mathematical texts from Cambridge University Press and relations to the honeycomb lattice examined by scholars at Yale University. Connections to higher-dimensional Coxeter groups and root systems appear in algebraic treatments linked to researchers at Institute for Advanced Study and in classification projects at Mathematical Sciences Research Institute. Dualities and tiling substitutions relate this lattice to Penrose-type tilings studied by mathematicians associated with University of London and Rutgers University.
Category:Lattice (group)