Hexagonal lattice on:
[Wikipedia]
[Google]
[Amazon]
The hexagonal lattice (sometimes called triangular lattice) is one of the five two-dimensional
The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices.
In nature,
Bravais lattice
In geometry and crystallography, a Bravais lattice, named after , is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
: \mathbf = n_1 \mathbf_1 + n_2 \mathbf_2 ...
types. The symmetry
Symmetry () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
category of the lattice is wallpaper group
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetry, symmetries in the pattern. Such patterns occur frequently in architecture a ...
p6m. The primitive translation vectors of the hexagonal lattice form an angle of 120° and are of equal lengths,
:
The reciprocal lattice
Reciprocal lattice is a concept associated with solids with translational symmetry which plays a major role in many areas such as X-ray and electron diffraction as well as the energies of electrons in a solid. It emerges from the Fourier tran ...
of the hexagonal lattice is a hexagonal lattice in reciprocal space with orientation changed by 90° and primitive lattice vectors of length
:
Honeycomb point set
The honeycomb point set is a special case of the hexagonal lattice with a two-atom basis. The centers of the hexagons of a honeycomb form a hexagonal lattice, and the honeycomb point set can be seen as the union of two offset hexagonal lattices.
In nature, carbon
Carbon () is a chemical element; it has chemical symbol, symbol C and atomic number 6. It is nonmetallic and tetravalence, tetravalent—meaning that its atoms are able to form up to four covalent bonds due to its valence shell exhibiting 4 ...
atoms of the two-dimensional material graphene
Graphene () is a carbon allotrope consisting of a Single-layer materials, single layer of atoms arranged in a hexagonal lattice, honeycomb planar nanostructure. The name "graphene" is derived from "graphite" and the suffix -ene, indicating ...
are arranged in a honeycomb point set.
Crystal classes
The ''hexagonal lattice'' class names,Schönflies notation
The Schoenflies (or Schönflies) notation, named after the German mathematician Arthur Moritz Schoenflies, is a notation primarily used to specify point groups in three dimensions. Because a point group alone is completely adequate to describe ...
, Hermann-Mauguin notation, orbifold notation
In geometry, orbifold notation (or orbifold signature) is a system, invented by the mathematician William Thurston and promoted by John Horton Conway, John Conway, for representing types of symmetry groups in two-dimensional spaces of constant curv ...
, Coxeter notation
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of a Coxeter group in a bracketed notation expressing the structure of a Coxeter-Dynkin diagram, ...
, and wallpaper groups
A wallpaper group (or plane symmetry group or plane crystallographic group) is a mathematical classification of a two-dimensional repetitive pattern, based on the symmetries in the pattern. Such patterns occur frequently in architecture and decorat ...
are listed in the table below.
See also
*Square lattice
In mathematics, the square lattice is a type of lattice in a two-dimensional Euclidean space. It is the two-dimensional version of the integer lattice, denoted as . It is one of the five types of two-dimensional lattices as classified by their ...
*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 or (as a Truncation (geometry), truncated triangular tiling ...
*Close-packing
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occ ...
*Centered hexagonal number
In mathematics and combinatorics, a centered hexagonal number, or centered hexagon number, is a centered polygonal number, centered figurate number that represents a hexagon with a dot in the center and all other dots surrounding the center dot ...
*Eisenstein integer
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the complex numbers of the form
: z = a + b\omega ,
where and are integers and
: \omega = \frac ...
*Voronoi diagram
In mathematics, a Voronoi diagram is a partition of a plane into regions close to each of a given set of objects. It can be classified also as a tessellation. In the simplest case, these objects are just finitely many points in the plane (calle ...
*Hermite constant
In mathematics, the Hermite constant, named after Charles Hermite, determines how long a shortest element of a lattice in Euclidean space can be.
The constant ''γn'' for integers ''n'' > 0 is defined as follows. For a lattice ''L'' in Euclidea ...
References
{{Commons category, position=left, Hexagonal lattices Lattice points Crystal systems