Nomenclature and generation of three-periodic nets: the vector method

Chung, Se-Hwa; Hahn, Th.; Klee, W. E.

doi:10.1107/s0108767384000088

Cited by 137 publications

(92 citation statements)

References 6 publications

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“…As a consequence, different non-isomorphic graphs may have the same quotient graph (Klee, 1987). A system of labels can however be assigned to the edges in such a way that it uniquely determines the infinite structure up to isomorphism via the so-called vector method (Chung et al, 1984). This opens the possibility of an algebraic representation of the quotient graph, which makes it possible to translate the process of comparing crystal structures into a symbolic computer language.…”

Section: Topology Of Crystal Structuresmentioning

confidence: 99%

Does mathematical crystallography still have a role in the XXI century?

Nespolo

2007

Acta Cryst Sect A

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Mathematical crystallography is the branch of crystallography dealing specifically with the fundamental properties of symmetry and periodicity of crystals, topological properties of crystal structures, twins, modular and modulated structures, polytypes and OD structures, as well as the symmetry aspects of phase transitions and physical properties of crystals. Mathematical crystallography has had its most evident success with the development of the theory of space groups at the end of the XIX century; since then, it has greatly enlarged its applications, but crystallographers are not always familiar with the developments that followed, partly because the applications sometimes require some additional background that the structural crystallographer does not always possess (as is the case, for example, in graph theory). The knowledge offered by mathematical crystallography is at present only partly mirrored in International Tables for Crystallography and is sometimes still enshrined in more specialist texts and publications. To cover this communication gap is one of the tasks of the IUCr Commission on Mathematical and Theoretical Crystallography (MaThCryst).

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Section: Topology Of Crystal Structuresmentioning

confidence: 99%

Does mathematical crystallography still have a role in the XXI century?

Nespolo

2007

Acta Cryst Sect A

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“…It is convenient to distinguish between abstract nets and their Euclidean embeddings (Chung et al, 1984). Specifically the vertices and edges of the graph correspond in this paper to nodes and kinks in an embedding.…”

Section: Minimal Nets and Their Embeddingsmentioning

confidence: 99%

“…Minimal nets are those that have the minimal number of vertices and edges in their repeat units. More specifically, their quotient graphs (Chung et al, 1984) have cyclomatic number, g, equal to d where d is the periodicity of the net. In this paper we are concerned exclusively with 3-periodic structures, so d = 3.…”

Section: Introductionmentioning

confidence: 99%

Minimal nets and minimal minimal surfaces

et al. 2013

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Author(s) of this paper may load this reprint on their own web site or institutional repository provided that this cover page is retained. Republication of this article or its storage in electronic databases other than as specified above is not permitted without prior permission in writing from the IUCr.For further information see http://journals.iucr.org/services/authorrights.html Acta Crystallographica Section A: Foundations of Crystallography covers theoretical and fundamental aspects of the structure of matter. The journal is the prime forum for research in diffraction physics and the theory of crystallographic structure determination by diffraction methods using X-rays, neutrons and electrons. The structures include periodic and aperiodic crystals, and non-periodic disordered materials, and the corresponding Bragg, satellite and diffuse scattering, thermal motion and symmetry aspects. Spatial resolutions range from the subatomic domain in charge-density studies to nanodimensional imperfections such as dislocations and twin walls. The chemistry encompasses metals, alloys, and inorganic, organic and biological materials. Structure prediction and properties such as the theory of phase transformations are also covered.Crystallography Journals Online is available from journals.iucr.org Acta Cryst. (2013 The 3-periodic nets of genus 3 ('minimal nets') are reviewed and their symmetries re-examined. Although they are all crystallographic, seven of the 15 only have maximum-symmetry embeddings if some links are allowed to have zero length. The connection between the minimal nets and the genus-3 zeromean-curvature surfaces ('minimal minimal' surfaces) is explored by determining the surface associated with a net that has a self-dual tiling. The fact that there are only five such surfaces but 15 minimal nets is rationalized by showing that all the minimal nets can serve as the labyrinth graph of one of the known minimal minimal surfaces.

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“…However, within the framework of this model, the immediate analysis of only local topology is possible. For the study of the global topological properties and also for the machine representation of an in®nite graph, it is convenient to use the method of reducing it to a ®nite`reduced' graph (RG) or so-called labeled quotient graph, as was suggested by Chung et al (1984). The operation of reducing can be visually represented by closing the edges of an in®nite graph, which are extended outside or are on the boundary of a unit cell, to translationally identical vertices, being inside the unit cell or on its boundary (Fig.…”

Section: Introductionmentioning

confidence: 99%

“…Hereinafter, we use the term subnet' to pay attention to a system of interatomic bonds and the term`sublattice' to emphasize that the topological properties are insigni®cant in such a context. According to Chung et al (1984), it is necessary to mark properly the RG vertices, which are incident to loops and multiple edges to store the information on the topology of these subnets. For this purpose, let us select in each subnet an origin atom being inside a given unit cell and assign to it the code (0, 0, 0).…”

Section: Introductionmentioning

confidence: 99%

Search for isotypism in crystal structures by means of the graph theory

Блатов

2000

Acta Cryst Sect A

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A method for the classi®cation of crystal structures of chemical compounds is proposed, which is based on the representation of the system of interatomic bonds in a crystal as a ®nite`reduced' graph. The program IsoTest is described, allowing one to ®nd automatically the topological similarity (isotypism) for large groups of stoichiometrically and structurally different compounds. The analysis of crystal structures of simple and double sulfates and binary inorganic compounds was carried out and numerous examples of topological isotypism of the representatives of these groups of substances were found. It is shown that in many cases the ionic sublattices, constructed according to one of the close packings, can be selected in the sulfate crystal structures.

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Nomenclature and generation of three-periodic nets: the vector method

Cited by 137 publications

References 6 publications

Does mathematical crystallography still have a role in the XXI century?

Does mathematical crystallography still have a role in the XXI century?

Minimal nets and minimal minimal surfaces

Search for isotypism in crystal structures by means of the graph theory

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