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Special Issue "Polyhedral Structures"

A special issue of Symmetry (ISSN 2073-8994).

Deadline for manuscript submissions: 30 April 2017

Special Issue Editor

Guest Editor
Prof. Dr. Egon Schulte

Northeastern University, Department of Mathematics, Boston, MA 02115, USA
Website | E-Mail
Phone: 617-373-5511
Interests: discrete and combinatorial geometry; combinatorics; group theory; graph theory

Special Issue Information

Dear Colleagues,

Since ancient times, mathematicians and scientists have been studying the geometry of polyhedra and polyhedral structures in ordinary Euclidean space. With the passage of time, various notions of polyhedra have attracted attention and have brought to light new exciting classes of symmetric structures, including the well-known Platonic and Archimedean solids, the Kepler-Poinsot star polyhedra, the Petrie-Coxeter polyhedra, and the Grünbaum-Dress polyhedra, as well as the more recently discovered chiral skeletal polyhedra and regular polygonal complexes. Over time we can observe a shift from the classical approach of viewing a polyhedron as a solid, to topological and algebraic approaches focussing on the underlying maps on surfaces, to graph-theoretical approaches highlighting the combinatorial incidence structures and featuring a polyhedron as a skeletal figure in space.

This Special Issue of Symmetry features articles about polyhedral structures, with symmetry as the unifying theme. We are soliciting contributions covering a broad range of topics including:  convex and non-convex polyhedra and higher-dimensional polytopes in spherical, euclidean, hyperbolic, or other spaces; skeletal polyhedral structures and their graphs; maps and polyhedra on surfaces of higher genus; abstract polyhedra and polytopes; polytopes, symmetry groups, and reflection groups; classification of polytopes by transitivity properties of symmetry groups; regular, chiral, and Archimedean polyhedra and polytopes; various classes of highly-symmetric polyhedra, such as vertex-, edge, or face-transitive polyhedra, regular-faced polyhedra, and equivelar maps or polyhedra; tessellations and space-fillers; polyhedra and crystallography; polyhedra in nature; polyhedra in art, design, ornament, and architecture; polyhedral models; and polyhedral design. 

Prof. Dr. Egon Schulte
Guest Editor

Submission

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. Papers will be published continuously (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as 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 refereed through a 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 1000 CHF (Swiss Francs).

Keywords

  • regular polyhedra and polytopes
  • symmetry groups and reflection groups
  • classification by symmetry
  • polyhedra and maps on surfaces
  • abstract polytopes
  • skeletal polyhedral structures and polyhedral graphs
  • polyhedral modeling of crystals
  • polyhedra in nature
  • polyhedral design

Published Papers (5 papers)

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Research

Open AccessArticle The Roundest Polyhedra with Symmetry Constraints
Symmetry 2017, 9(3), 41; doi:10.3390/sym9030041
Received: 5 December 2016 / Revised: 3 March 2017 / Accepted: 8 March 2017 / Published: 15 March 2017
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Abstract
Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the
[...] Read more.
Amongst the convex polyhedra with n faces circumscribed about the unit sphere, which has the minimum surface area? This is the isoperimetric problem in discrete geometry which is addressed in this study. The solution of this problem represents the closest approximation of the sphere, i.e., the roundest polyhedra. A new numerical optimization method developed previously by the authors has been applied to optimize polyhedra to best approximate a sphere if tetrahedral, octahedral, or icosahedral symmetry constraints are applied. In addition to evidence provided for various cases of face numbers, potentially optimal polyhedra are also shown for n up to 132. Full article
(This article belongs to the Special Issue Polyhedral Structures)
Figures

Figure 1

Open AccessArticle Aesthetic Patterns with Symmetries of the Regular Polyhedron
Symmetry 2017, 9(2), 21; doi:10.3390/sym9020021
Received: 14 December 2016 / Revised: 18 January 2017 / Accepted: 22 January 2017 / Published: 3 February 2017
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Abstract A fast algorithm is established to transform points of the unit sphere into fundamental region symmetrically. With the resulting algorithm, a flexible form of invariant mappings is achieved to generate aesthetic patterns with symmetries of the regular polyhedra. Full article
(This article belongs to the Special Issue Polyhedral Structures)
Figures

Open AccessArticle On Center, Periphery and Average Eccentricity for the Convex Polytopes
Symmetry 2016, 8(12), 145; doi:10.3390/sym8120145
Received: 3 November 2016 / Revised: 22 November 2016 / Accepted: 24 November 2016 / Published: 2 December 2016
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Abstract
A vertex v is a peripheral vertex in G if its eccentricity is equal to its diameter, and periphery P(G) is a subgraph of G induced by its peripheral vertices. Further, a vertex v in G is a central vertex
[...] Read more.
A vertex v is a peripheral vertex in G if its eccentricity is equal to its diameter, and periphery P ( G ) is a subgraph of G induced by its peripheral vertices. Further, a vertex v in G is a central vertex if e ( v ) = r a d ( G ) , and the subgraph of G induced by its central vertices is called center C ( G ) of G . Average eccentricity is the sum of eccentricities of all of the vertices in a graph divided by the total number of vertices, i.e., a v e c ( G ) = { 1 n e G ( u ) ; u V ( G ) } . If every vertex in G is central vertex, then C ( G ) = G , and hence, G is self-centered. In this report, we find the center, periphery and average eccentricity for the convex polytopes. Full article
(This article belongs to the Special Issue Polyhedral Structures)
Figures

Figure 1

Open AccessArticle Regular and Chiral Polyhedra in Euclidean Nets
Symmetry 2016, 8(11), 115; doi:10.3390/sym8110115
Received: 14 September 2016 / Revised: 21 October 2016 / Accepted: 23 October 2016 / Published: 28 October 2016
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Abstract We enumerate the regular and chiral polyhedra (in the sense of Grünbaum’s skeletal approach) whose vertex and edge sets are a subset of those of the primitive cubic lattice, the face-centred cubic lattice, or the body-centred cubic lattice. Full article
(This article belongs to the Special Issue Polyhedral Structures)
Figures

Figure 1

Open AccessArticle Decoration of the Truncated Tetrahedron—An Archimedean Polyhedron—To Produce a New Class of Convex Equilateral Polyhedra with Tetrahedral Symmetry
Symmetry 2016, 8(8), 82; doi:10.3390/sym8080082
Received: 27 July 2016 / Revised: 12 August 2016 / Accepted: 16 August 2016 / Published: 20 August 2016
PDF Full-text (7222 KB) | HTML Full-text | XML Full-text | Supplementary Files
Abstract
The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a
[...] Read more.
The Goldberg construction of symmetric cages involves pasting a patch cut out of a regular tiling onto the faces of a Platonic host polyhedron, resulting in a cage with the same symmetry as the host. For example, cutting equilateral triangular patches from a 6.6.6 tiling of hexagons and pasting them onto the full triangular faces of an icosahedron produces icosahedral fullerene cages. Here we show that pasting cutouts from a 6.6.6 tiling onto the full hexagonal and triangular faces of an Archimedean host polyhedron, the truncated tetrahedron, produces two series of tetrahedral (Td) fullerene cages. Cages in the first series have 28n2 vertices (n ≥ 1). Cages in the second (leapfrog) series have 3 × 28n2. We can transform all of the cages of the first series and the smallest cage of the second series into geometrically convex equilateral polyhedra. With tetrahedral (Td) symmetry, these new polyhedra constitute a new class of “convex equilateral polyhedra with polyhedral symmetry”. We also show that none of the other Archimedean polyhedra, six with octahedral symmetry and six with icosahedral, can host full-face cutouts from regular tilings to produce cages with the host’s polyhedral symmetry. Full article
(This article belongs to the Special Issue Polyhedral Structures)
Figures

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