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Symmetry
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Mathematical Crystallography
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Mathematical Crystallography

A special issue of Symmetry (ISSN 2073-8994). This special issue belongs to the section "Mathematics".

Deadline for manuscript submissions: closed (15 January 2018) | Viewed by 20005

Special Issue Editor

Prof. Sergey V. Krivovichev

E-Mail Website
Guest Editor
Department of crystallography, Institute of Earth Sciences, Saint-Petersburg State University, 199155 Saint-Petersburg, Russia
Interests: mineralogy; crystallography; complexity; crystal chemistry; nuclear chemistry
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Despite the enormous success of experimental crystallography achieved in 20th century, the very problem of the emergence of order and symmetry in discrete structures is far from being completely understood. This Special Issue invites contributions on various mathematical aspects of modern crystallography, starting from diffraction theory and disorder and ending at deeply mathematical problems of symmetry, sphere packings, polyhedra and discrete point sets.

Prof. Dr. Sergey V. Krivovichev
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Symmetry is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2400 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • crystals
  • crystal structure
  • symmetry
  • long-range order
  • sphere packings
  • discrete point systems
  • polyhedral arrangements
  • lattices
  • space groups
  • quasicrystals

Published Papers (5 papers)

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Research

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28 pages, 636 KiB  
Article
Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face
by Nikolai Erokhovets
Symmetry 2018, 10(3), 67; https://doi.org/10.3390/sym10030067 - 15 Mar 2018
Cited by 8 | Viewed by 3762
Abstract
A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed [...] Read more.
A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed numbers of k-gons, k ≥ 7 . Any simple polytope with only 5-, 6- and at most one 7-gon is Pogorelov. For any other prescribed numbers of k-gons, k ≥ 7 , we give an explicit construction of a Pogorelov and a non-Pogorelov polytope. Any Pogorelov polytope different from k-barrels (also known as Löbel polytopes, whose graphs are biladders on 2 k vertices) can be constructed from the 5- or the 6-barrel by cutting off pairs of adjacent edges and connected sums with the 5-barrel along a 5-gon with the intermediate polytopes being Pogorelov. For fullerenes, there is a stronger result. Any fullerene different from the 5-barrel and the ( 5 , 0 ) -nanotubes can be constructed by only cutting off adjacent edges from the 6-barrel with all the intermediate polytopes having 5-, 6- and at most one additional 7-gon adjacent to a 5-gon. This result cannot be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the 5-barrel and 3 new operations, which are compositions of cutting off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons. Full article
(This article belongs to the Special Issue Mathematical Crystallography)
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13 pages, 1523 KiB  
Article
Symmetry and Topology: The 11 Uninodal Planar Nets Revisited
by Jean-Guillaume Eon
Symmetry 2018, 10(2), 35; https://doi.org/10.3390/sym10020035 - 24 Jan 2018
Cited by 4 | Viewed by 5057
Abstract
A description of the 11 well-known uninodal planar nets is given by Cayley color graphs or alternative Cayley color graphs of plane groups. By applying methods from topological graph theory, the nets are derived from the bouquet B n with rotations mostly as [...] Read more.
A description of the 11 well-known uninodal planar nets is given by Cayley color graphs or alternative Cayley color graphs of plane groups. By applying methods from topological graph theory, the nets are derived from the bouquet B n with rotations mostly as voltages. It thus appears that translation, as a symmetry operation in these nets, is no more fundamental than rotations. Full article
(This article belongs to the Special Issue Mathematical Crystallography)
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203 KiB  
Article
The Exact Evaluation of Some New Lattice Sums
by I. J. Zucker
Symmetry 2017, 9(12), 314; https://doi.org/10.3390/sym9120314 - 11 Dec 2017
Cited by 6 | Viewed by 3045
Abstract
New q-series in the spirit of Jacobi have been found in a publication first published in 1884 written in Russian and translated into English in 1928. This work was found by chance and appears to be almost totally unknown. From these entirely [...] Read more.
New q-series in the spirit of Jacobi have been found in a publication first published in 1884 written in Russian and translated into English in 1928. This work was found by chance and appears to be almost totally unknown. From these entirely new q-series, fresh lattice sums have been discovered and are presented here. Full article
(This article belongs to the Special Issue Mathematical Crystallography)

Review

Jump to: Research

17 pages, 778 KiB  
Review
The Local Theory for Regular Systems in the Context of t-Bonded Sets
by Mikhail Bouniaev and Nikolay Dolbilin
Symmetry 2018, 10(5), 159; https://doi.org/10.3390/sym10050159 - 14 May 2018
Viewed by 2678
Abstract
The main goal of the local theory for crystals developed in the last quarter of the 20th Century by a geometry group of Delone (Delaunay) at the Steklov Mathematical Institute is to find and prove the correct statements rigorously explaining why the crystalline [...] Read more.
The main goal of the local theory for crystals developed in the last quarter of the 20th Century by a geometry group of Delone (Delaunay) at the Steklov Mathematical Institute is to find and prove the correct statements rigorously explaining why the crystalline structure follows from the pair-wise identity of local arrangements around each atom. Originally, the local theory for regular and multiregular systems was developed with the assumption that all point sets under consideration are ( r , R ) -systems or, in other words, Delone sets of type ( r , R ) in d-dimensional Euclidean space. In this paper, we will review the recent results of the local theory for a wider class of point sets compared with the Delone sets. We call them t-bonded sets. This theory, in particular, might provide new insight into the case for which the atomic structure of matter is a Delone set of a “microporous” character, i.e., a set that contains relatively large cavities free from points of the set. Full article
(This article belongs to the Special Issue Mathematical Crystallography)
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46 pages, 5676 KiB  
Review
Towards Generalized Noise-Level Dependent Crystallographic Symmetry Classifications of More or Less Periodic Crystal Patterns
by Peter Moeck
Symmetry 2018, 10(5), 133; https://doi.org/10.3390/sym10050133 - 25 Apr 2018
Cited by 15 | Viewed by 4414
Abstract
Geometric Akaike Information Criteria (G-AICs) for generalized noise-level dependent crystallographic symmetry classifications of two-dimensional (2D) images that are more or less periodic in either two or one dimensions as well as Akaike weights for multi-model inferences and predictions are reviewed. Such novel classifications [...] Read more.
Geometric Akaike Information Criteria (G-AICs) for generalized noise-level dependent crystallographic symmetry classifications of two-dimensional (2D) images that are more or less periodic in either two or one dimensions as well as Akaike weights for multi-model inferences and predictions are reviewed. Such novel classifications do not refer to a single crystallographic symmetry class exclusively in a qualitative and definitive way. Instead, they are quantitative, spread over a range of crystallographic symmetry classes, and provide opportunities for inferences from all classes (within the range) simultaneously. The novel classifications are based on information theory and depend only on information that has been extracted from the images themselves by means of maximal likelihood approaches so that these classifications are objective. This is in stark contrast to the common practice whereby arbitrarily set thresholds or null hypothesis tests are employed to force crystallographic symmetry classifications into apparently definitive/exclusive states, while the geometric feature extraction results on which they depend are never definitive in the presence of generalized noise, i.e., in all real-world applications. Thus, there is unnecessary subjectivity in the currently practiced ways of making crystallographic symmetry classifications, which can be overcome by the approach outlined in this review. Full article
(This article belongs to the Special Issue Mathematical Crystallography)
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