On the Bravais types of lattices

Delone, B.; Galiulin, R. V.; Штогрин, Михаил Иванович

doi:10.1007/bf01084661

Cited by 17 publications

(19 citation statements)

References 15 publications

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“…Delone et al (1975) state 'The Selling parameters are geometrically fully homogeneous'. Delone et al (1975) state 'The Selling parameters are geometrically fully homogeneous'.…”

Section: The Space Smentioning

confidence: 99%

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A space for lattice representation and clustering

2019

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Algorithms for quantifying the differences between two lattices are used for Bravais lattice determination, database lookup for unit cells to select candidates for molecular replacement, and recently for clustering to group together images from serial crystallography. It is particularly desirable for the differences between lattices to be computed as a perturbation-stable metric, i.e. as distances that satisfy the triangle inequality, so that standard tree-based nearest-neighbor algorithms can be used, and for which small changes in the lattices involved produce small changes in the distances computed. A perturbation-stable metric space related to the reduction algorithm of Selling and to the Bravais lattice determination methods of Delone is described. Two ways of representing the space, as six-dimensional real vectors or equivalently as three-dimensional complex vectors, are presented and applications of these metrics are discussed. (Note: in his later publications, Boris Delaunay used the Russian version of his surname, Delone.)

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“…Delone et al (1975) state 'The Selling parameters are geometrically fully homogeneous'. Delone et al (1975) state 'The Selling parameters are geometrically fully homogeneous'.…”

Section: The Space Smentioning

confidence: 99%

“…; s 6 , the components of C 3 are the pairs of 'opposite' (Delone et al, 1975) scalars. C 3 has advantages for understanding some of the properties of the space.…”

Section: The Space Cmentioning

confidence: 99%

A space for lattice representation and clustering

2019

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“…This is only a formal labeling; associated with each pair of vertices, the edge between them is labeled with the dot product of the two vectors ending at those vertices. Delone et al (1975) state 'Select any positive parameter of the tetrahedron and subtract it from the parameter standing on the opposite edge of the tetrahedron (the tetrahedron is at all times thought of as spatial), add it to the parameters standing on the remaining four edges, interchange the places of the obtained parameters on two of these four edges, converging to one of the ends of the original edge (it makes no difference to which), and, finally, change the sign of the positive parameter itself being considered. '…”

Section: The Tetrahedronmentioning

confidence: 99%

“…Taking squares of these lengths gives a seven-vector defined by Delone et al (1975) in a space we call D 7 for Delone sevenspace.…”

Section: A5 the Seven-dimensional Delone Space Dmentioning

confidence: 99%

Selling reduction versus Niggli reduction for crystallographic lattices

2019

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“…Both the combinatorial-topological structure and symmetry of a crystallographic plane are described in terms of planigons and symmetry. Closely connected with the theory of planigons is the so-called "local theorem" [7], on the basis of which the consistent geometrical crystallography may be developed [8]. Figure 1(a) shows as an example one of four possible tessellations for the p6 symmetry group.…”

Section: Introductionmentioning

confidence: 99%

Planigon tessellation cellular automata

Коробов

1999

Complexity

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Cellular automata (CA) do not find as wide a use in chemical complexity determined by a crystal structure as in homogeneous reacting systems. A reason for this is discussed, and a superposition of planigons and parallelogons or Wigner‐Seitz tessellations is suggested that is derived from a crystallographic plane and is capable of representing chemical complexity at a single crystal face in terms of CA. The interplay between growth and form of two‐dimensional negative crystals is discussed in these terms as a particular example. © 1999 John Wiley & Sons, Inc.

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On the Bravais types of lattices

Cited by 17 publications

References 15 publications

A space for lattice representation and clustering

A space for lattice representation and clustering

Selling reduction versus Niggli reduction for crystallographic lattices

Planigon tessellation cellular automata

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