Special Issue "Polyhedra"

Guest Editor
Prof. Dr. Egon Schulte
Northeastern University, Department of Mathematics, Boston, MA 02115, USA
Website: http://www.math.neu.edu/people/profile/egon-schulte
E-Mail: schulte@neu.edu
Interests: discrete and combinatorial geometry; combinatorics; group theory; graph theory

Dear Colleagues,

The study of polyhedra and symmetry has a long and fascinating history tracing back to the early days of geometry. With the passage of time, various notions of polyhedra have attracted attention and have brought to light new exciting classes of regular or other symmetric polyhedra including well-known figures such as the Platonic solids, Kepler-Poinsot star polyhedra, and Petrie-Coxeter sponge polyhedra, as well as the more recently discovered new regular or chiral skeletal polyhedra. This flexibility of the concept proves an important point --- polyhedra and symmetry have shown an enormous potential for revival! One explanation for this is the appearance of symmetric polyhedra in many contexts that a priori seem to have little apparent relation to symmetry, such as the occurrence of many figures in nature as crystals. In addition, their internal beauty appeals to the artistic senses and sparks the desire for a rigorous mathematical analysis and understanding of the figures themselves, as well as of their relationships with other areas of science.

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

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 quarterly 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 300 CHF (Swiss Francs). English correction and/or formatting fees of 250 CHF (Swiss Francs) will be charged in certain cases for those articles accepted for publication that require extensive additional formatting and/or English corrections.

Egon Schulte and Jörg M. Wills Article: Convex-Faced Combinatorially Regular Polyhedra of Small Genus Symmetry 2012, 4(1), 1-14; doi:10.3390/sym4010001 Received: 28 November 2011; in revised form: 15 December 2011 / Accepted: 19 December 2011 / Published: 28 December 2011 Show/Hide Abstract | Download PDF Full-text (271 KB) Abstract: Combinatorially regular polyhedra are polyhedral realizations (embeddings) in Euclidean 3-space E3 of regular maps on (orientable) closed compact surfaces. They are close analogues of the Platonic solids. A surface of genus g ≥ 2 admits only finitely many regular maps, and generally only a small number of them can be realized as polyhedra with convex faces. When the genus g is small, meaning that g is in the historically motivated range 2 ≤ g ≤ 6, only eight regular maps of genus g are known to have polyhedral realizations, two discovered quite recently. These include spectacular convex-faced polyhedra realizing famous maps of Klein, Fricke, Dyck, and Coxeter. We provide supporting evidence that this list is complete; in other words, we strongly conjecture that in addition to those eight there are no other regular maps of genus g, with 2 ≤ g ≤ 6, admitting realizations as convex-faced polyhedra in E3. For all admissible maps in this range, save Gordan’s map of genus 4, and its dual, we rule out realizability by a polyhedron in E3.

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Gabe Cunningham Article: Self-Dual, Self-Petrie Covers of Regular Polyhedra Symmetry 2012, 4(1), 208-218; doi:10.3390/sym4010208 Received: 17 January 2012; in revised form: 21 February 2012 / Accepted: 23 February 2012 / Published: 27 February 2012 Show/Hide Abstract | Download PDF Full-text (234 KB) Abstract: The well-known duality and Petrie duality operations on maps have natural analogs for abstract polyhedra. Regular polyhedra that are invariant under both operations have a high degree of both “external” and “internal” symmetry. The mixing operation provides a natural way to build the minimal common cover of two polyhedra, and by mixing a regular polyhedron with its five other images under the duality operations, we are able to construct the minimal self-dual, self-Petrie cover of a regular polyhedron. Determining the full structure of these covers is challenging and generally requires that we use some of the standard algorithms in combinatorial group theory. However, we are able to develop criteria that sometimes yield the full structure without explicit calculations. Using these criteria and other interesting methods, we then calculate the size of the self-dual, self-Petrie covers of several polyhedra, including the regular convex polyhedra.

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Steve Wilson Article: Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators Symmetry 2012, 4(2), 265-275; doi:10.3390/sym4020265 Received: 29 January 2012; in revised form: 26 March 2012 / Accepted: 5 April 2012 / Published: 16 April 2012 Show/Hide Abstract | Download PDF Full-text (262 KB) Abstract: This paper introduces the idea of a maniplex, a common generalization of map and of polytope. The paper then discusses operators, orientability, symmetry and the action of the symmetry group.

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Type of Paper: Article
Title: Sphere-covering Fractal Curves Using Polyhedral Initiators - A Method for Classification and Generation
Author: Jeffrey Ventrella
Affiliation: Visual Music Systems, Boston, MA, USA; Email: jeffreyventrella@gmail.com
Abstract: Space-filling curves can be used to model river basins, vascular systems, and solutions to the traveling salesman problem. They also generate a variety of aesthetic forms. A plane-filling curve maps a 1D line to a 2D planar region. Can a spherical great circle likewise be mapped to the entire sphere? This paper presents an extension of a scheme for generating and classifying space-filling curves, from lines in the plane to great circles on the sphere, whose positive curvature imposes unique grid contraints. All Platonic solids except for the dodecahedron admit "pertiling" (recursive self-similar tiling). The set of all possible initiators for spherical Koch-constructed (edge-replacement) closed curves are presented, with comparative analysis.

Type of Paper: Article
Title: Non-crystallographic Symmetry Based on Discrete Packing Space Model
Author: Valery Rau, et al.
Affiliation: Vladimir State Humanitarian University, Russia; E-Mail: vgrau@mail.ru
Abstract: Isomorphism of finite groups of symmetry and non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups) is considered. For the first time it is implemented in the form of structures groups of colored partitions in discrete periodic packing spaces introduced above. On the one hand, such an approach establishes computer design of abstract groups of symmetry; on the other hand it shows us that any of periodic crystal structure can be represented as element of a particular subgroup of transformations substitution. Process of discrete strain transformations (phase transitions) in real structures is demonstrated for the first time using finite exhaustive search of sub-lattices of packing spaces.