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The regular skew polyhedron onto which the Laves graph can be inscribed. The edges of the Laves graph are diagonals of some of the squares of this polyhedral surface. As Coxeter (1955) describes, the vertices of the Laves graph can be defined by selecting one out of every eight points in the three-dimensional integer lattice, and forming their nearest neighbor graph.
A lattice in the sense of a 3-dimensional array of regularly spaced points coinciding with e.g. the atom or molecule positions in a crystal, or more generally, the orbit of a group action under translational symmetry, is a translation of the translation lattice: a coset, which need not contain the origin, and therefore need not be a lattice in ...
A common type of lattice graph (known under different names, such as grid graph or square grid graph) is the graph whose vertices correspond to the points in the plane with integer coordinates, x-coordinates being in the range 1, ..., n, y-coordinates being in the range 1, ..., m, and two vertices being connected by an edge whenever the corresponding points are at distance 1.
In differential geometry, a triply periodic minimal surface (TPMS) is a minimal surface in that is invariant under a rank-3 lattice of translations. These surfaces have the symmetries of a crystallographic group. Numerous examples are known with cubic, tetragonal, rhombohedral, and orthorhombic symmetries.
Applications in macroscopic engineering have been suggested, building quasi-crystal-like large scale engineering structures, which could have interesting physical properties. Also, aperiodic tiling lattice structures may be used instead of isogrid or honeycomb patterns. None of these seem to have been put to use in practice. [67]
In signal processing, multidimensional discrete convolution refers to the mathematical operation between two functions f and g on an n-dimensional lattice that produces a third function, also of n-dimensions. Multidimensional discrete convolution is the discrete analog of the multidimensional convolution of functions on Euclidean space.
The seven lattice systems and their Bravais lattices in three dimensions. In geometry and crystallography, a Bravais lattice, named after Auguste Bravais (), [1] is an infinite array of discrete points generated by a set of discrete translation operations described in three dimensional space by
In the theory of lattices, the dual lattice is a construction analogous to that of a dual vector space.In certain respects, the geometry of the dual lattice of a lattice is the reciprocal of the geometry of , a perspective which underlies many of its uses.