Symmetry in Discrete and Combinatorial Geometry

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

Deadline for manuscript submissions: 31 December 2024 | Viewed by 6142

Special Issue Editors

Department of Mathematics, Northeastern University, Boston, MA, USA
Interests: discrete and combinatorial geometry; combinatorics; group theory
Special Issues, Collections and Topics in MDPI journals
Mathematical Institute, Leiden University, 2333 CA Leiden, The Netherlands
Interests: algebraic geometry; derived algebraic geometry; combinatorial algebraic geometry; number theory

Special Issue Information

Dear Colleagues,

Symmetry is a frequently recurring theme in mathematics, in science and nature, and in the arts. In mathematics, its most familiar manifestation appears in geometry, most notably discrete and combinatorial geometry, and in closely related areas. Highly-symmetric geometric figures have been studied since the early days of geometry. In fact, historically, a fair amount of geometry has been largely inspired by them. Apart from appealing to one's artistic sense, their inherent beauty invites rigorous mathematical analysis and discovery of the figures themselves, as well as of their relationships with other sciences.

This Special Issue of Symmetry features articles about geometric, combinatorial or algebraic symmetries in discrete objects from discrete and combinatorial geometry. We are soliciting contributions covering a broad range of topics related to symmetry, including convex and nonconvex polyhedra in spherical, Euclidean, hyperbolic, or other spaces; convex polytopes; tiling and space-fillers; polyhedra and crystallography; skeletal polyhedra; crystal nets; rigidity of frameworks; abstract polytopes and C-groups; maps on surfaces; geometric graphs; point configurations; Delone sets; aperiodic structures; packing and covering; sphere packings; soft packings, nested clusters, and condensed matter; reflection groups and Coxeter groups; polyhedral models; and polyhedra in art, design, and architecture.

Prof. Dr. Egon Schulte
Dr. Márton Hablicsek
Guest Editors

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

  • polyhedra and polytopes
  • polyhedra and crystallography,crystal nets, polyhedral graphs, and skeletal polyhedral structures
  • polyhedra and maps on surfaces
  • tiling, packing, and covering
  • soft packings, nested clusters, and condensed matter
  • symmetry groups and reflection groups
  • discrete point sets
  • abstract polytopes
  • polyhedral design

Published Papers (2 papers)

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Research

19 pages, 6858 KiB  
Article
The Symmetry and Topology of Finite and Periodic Graphs and Their Embeddings in Three-Dimensional Euclidean Space
by Michael O’Keeffe and Michael M. J. Treacy
Symmetry 2022, 14(4), 822; https://0-doi-org.brum.beds.ac.uk/10.3390/sym14040822 - 14 Apr 2022
Cited by 8 | Viewed by 2624
Abstract
We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. [...] Read more.
We make the case for the universal use of the Hermann-Mauguin (international) notation for the description of rigid-body symmetries in Euclidean space. We emphasize the importance of distinguishing between graphs and their embeddings and provide examples of 0-, 1-, 2-, and 3-periodic structures. Embeddings of graphs are given as piecewise linear with finite, non-intersecting edges. We call attention to problems of conflicting terminology when disciplines such as materials chemistry and mathematics collide. Full article
(This article belongs to the Special Issue Symmetry in Discrete and Combinatorial Geometry)
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18 pages, 856 KiB  
Article
Nested Polyhedra and Indices of Orbits of Coxeter Groups of Non-Crystallographic Type
by Mariia Myronova, Jiří Patera and Marzena Szajewska
Symmetry 2020, 12(10), 1737; https://0-doi-org.brum.beds.ac.uk/10.3390/sym12101737 - 20 Oct 2020
Cited by 1 | Viewed by 2196
Abstract
The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and [...] Read more.
The invariants of finite-dimensional representations of simple Lie algebras, such as even-degree indices and anomaly numbers, are considered in the context of the non-crystallographic finite reflection groups H2, H3 and H4. Using a representation-orbit replacement, the definitions and properties of the indices are formulated for individual orbits of the examined groups. The indices of orders two and four of the tensor product of k orbits are determined. Using the branching rules for the non-crystallographic Coxeter groups, the embedding index is defined similarly to the Dynkin index of a representation. Moreover, since the definition of the indices can be applied to any orbit of non-crystallographic type, the algorithm allowing to search for the orbits of smaller radii contained within any considered one is presented for the Coxeter groups H2 and H3. The geometrical structures of nested polytopes are exemplified. Full article
(This article belongs to the Special Issue Symmetry in Discrete and Combinatorial Geometry)
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