Symmetry and Duality

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

Deadline for manuscript submissions: closed (30 September 2015) | Viewed by 34471

Special Issue Editor


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Guest Editor
Mathematics Department, Worcester Polytechnic Institute, Worcester, MA 01609-2280, USA
Interests: combinatorics; rigiditiy of structures; geometric foundations of computer aided design; symmetry and duality; the history and philosophy of mathematics
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Special Issue Information

Dear Colleagues,

Symmetry and duality has a long history starting with the platonic solids where duality interchanges faces and vertices. Maxwell's reciprocal figures point out a duality between kinematics and statics. Polarity is another well-studied form of duality of geometric objects. For graphs embedded on the sphere, we have the Petrie dual, as well as the antipodal dual. Self-dual planar graphs and tilings have been classified by their symmetry group. On the sphere, there are no non-trivial regular graphs that are both self-dual and self-Petrie, but on the surfaces of higher genus, they do exist.

We want to collect the results and applications of various forms of duality under various forms of generalizations or relaxations. For example, dual Eulerian graphs (an Euler circuit in the graph is also a Euler circuit in the geometric dual) have been defined and studied in the context of the silicon optimization of CMOS layouts. Relatively new are partial and twisted dualities, with applications to knots. Maxwell's reciprocal figures found new applications in the study of rigidity and motions under imposed symmetry conditions.

Prof. Brigitte Servatius
Guest Editor

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Keywords

  • planar graphs
  • geometric and topological aspects of graph theory
  • graphs and abstract algebra
  • graph representations
  • convex and discrete geometry
  • polytopes and polyhedral
  • geometric graph theory
  • reflection and coxeter groups
  • combinatorial aspects of groups and algebras

Published Papers (7 papers)

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Research

303 KiB  
Article
E-Polytopes in Picard Groups of Smooth Rational Surfaces
by Jae-Hyouk Lee and YongJoo Shin
Symmetry 2016, 8(4), 27; https://doi.org/10.3390/sym8040027 - 20 Apr 2016
Cited by 2 | Viewed by 3985
Abstract
In this article, we introduce special divisors (root, line, ruling, exceptional system and rational quartic) in smooth rational surfaces and study their correspondences to subpolytopes in Gosset polytopes k 21 . We also show that the sets of rulings and exceptional systems correspond [...] Read more.
In this article, we introduce special divisors (root, line, ruling, exceptional system and rational quartic) in smooth rational surfaces and study their correspondences to subpolytopes in Gosset polytopes k 21 . We also show that the sets of rulings and exceptional systems correspond equivariantly to the vertices of 2 k 1 and 1 k 2 via E-type Weyl action. Full article
(This article belongs to the Special Issue Symmetry and Duality)
430 KiB  
Article
Dual Pairs of Holomorphic Representations of Lie Groups from a Vector-Coherent-State Perspective
by David J. Rowe and Joe Repka
Symmetry 2016, 8(3), 12; https://doi.org/10.3390/sym8030012 - 16 Mar 2016
Cited by 1 | Viewed by 3703
Abstract
It is shown that, for both compact and non-compact Lie groups, vector-coherent-state methods provide straightforward derivations of holomorphic representations on symmetric spaces. Complementary vector-coherent-state methods are introduced to derive pairs of holomorphic representations which are bi-orthogonal duals of each other with respect to [...] Read more.
It is shown that, for both compact and non-compact Lie groups, vector-coherent-state methods provide straightforward derivations of holomorphic representations on symmetric spaces. Complementary vector-coherent-state methods are introduced to derive pairs of holomorphic representations which are bi-orthogonal duals of each other with respect to a simple Bargmann inner product. It is then shown that the dual of a standard holomorphic representation has an integral expression for its inner product, with a Bargmann measure and a simply-defined kernel, which is not restricted to discrete-series representations. Dual pairs of holomorphic representations also provide practical ways to construct orthonormal bases for unitary irreps which bypass the need for evaluating the integral expressions for their inner products. This leads to practical algorithms for the application of holomorphic representations to model problems with dynamical symmetries in physics. Full article
(This article belongs to the Special Issue Symmetry and Duality)
1034 KiB  
Article
Duality in Geometric Graphs: Vector Graphs, Kirchhoff Graphs and Maxwell Reciprocal Figures
by Tyler Reese, Randy Paffenroth and Joseph D. Fehribach
Symmetry 2016, 8(3), 9; https://doi.org/10.3390/sym8030009 - 29 Feb 2016
Cited by 3 | Viewed by 5685
Abstract
We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing [...] Read more.
We compare two mathematical theories that address duality between cycles and vertex-cuts of graphs in geometric settings. First, we propose a rigorous definition of a new type of graph, vector graphs. The special case of R2-vector graphs matches the intuitive notion of drawing graphs with edges taken as vectors. This leads to a discussion of Kirchhoff graphs, as originally presented by Fehribach, which can be defined independent of any matrix relations. In particular, we present simple cases in which vector graphs are guaranteed to be Kirchhoff or non-Kirchhoff. Next, we review Maxwell’s method of drawing reciprocal figures as he presented in 1864, using modern mathematical language. We then demonstrate cases in which R2-vector graphs defined from Maxwell reciprocals are “dual” Kirchhoff graphs. Given an example in which Maxwell’s theories are not sufficient to define vector graphs, we begin to explore other methods of developing dual Kirchhoff graphs. Full article
(This article belongs to the Special Issue Symmetry and Duality)
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251 KiB  
Article
Petrie Duality and the Anstee–Robertson Graph
by Gareth A. Jones and Matan Ziv-Av
Symmetry 2015, 7(4), 2206-2223; https://doi.org/10.3390/sym7042206 - 21 Dec 2015
Cited by 3 | Viewed by 4230
Abstract
We define the operation of Petrie duality for maps, describing its general properties both geometrically and algebraically. We give a number of examples and applications, including the construction of a pair of regular maps, one orientable of genus 17, the other non-orientable of [...] Read more.
We define the operation of Petrie duality for maps, describing its general properties both geometrically and algebraically. We give a number of examples and applications, including the construction of a pair of regular maps, one orientable of genus 17, the other non-orientable of genus 52, which embed the 40-vertex cage of valency 6 and girth 5 discovered independently by Robertson and Anstee. We prove that this map (discovered by Evans) and its Petrie dual are the only regular embeddings of this graph, together with a similar result for a graph of order 40, valency 6 and girth 3 with the same automorphism group. Full article
(This article belongs to the Special Issue Symmetry and Duality)
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271 KiB  
Article
Three Duality Symmetries between Photons and Cosmic String Loops, and Macro and Micro Black Holes
by David Jou, Michele Sciacca and Maria Stella Mongiovì
Symmetry 2015, 7(4), 2134-2149; https://doi.org/10.3390/sym7042134 - 17 Nov 2015
Cited by 1 | Viewed by 4352
Abstract
We present a review of two thermal duality symmetries between two different kinds of systems: photons and cosmic string loops, and macro black holes and micro black holes, respectively. It also follows a third joint duality symmetry amongst them through thermal equilibrium and [...] Read more.
We present a review of two thermal duality symmetries between two different kinds of systems: photons and cosmic string loops, and macro black holes and micro black holes, respectively. It also follows a third joint duality symmetry amongst them through thermal equilibrium and stability between macro black holes and photon gas, and micro black holes and string loop gas, respectively. The possible cosmological consequences of these symmetries are discussed. Full article
(This article belongs to the Special Issue Symmetry and Duality)
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1942 KiB  
Article
Similarity and a Duality for Fullerenes
by Jennifer J. Edmond and Jack E. Graver
Symmetry 2015, 7(4), 2047-2061; https://doi.org/10.3390/sym7042047 - 6 Nov 2015
Viewed by 4572
Abstract
Fullerenes are molecules of carbon that are modeled by trivalent plane graphs with only pentagonal and hexagonal faces. Scaling up a fullerene gives a notion of similarity, and fullerenes are partitioned into similarity classes. In this expository article, we illustrate how the values [...] Read more.
Fullerenes are molecules of carbon that are modeled by trivalent plane graphs with only pentagonal and hexagonal faces. Scaling up a fullerene gives a notion of similarity, and fullerenes are partitioned into similarity classes. In this expository article, we illustrate how the values of two important fullerene parameters can be deduced for all fullerenes in a similarity class by computing the values of these parameters for just the three smallest representatives of that class. In addition, it turns out that there is a natural duality theory for similarity classes of fullerenes based on one of the most important fullerene construction techniques: leapfrog construction. The literature on fullerenes is very extensive, and since this is a general interest journal, we will summarize and illustrate the fundamental results that we will need to develop similarity and this duality. Full article
(This article belongs to the Special Issue Symmetry and Duality)
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375 KiB  
Article
Mirror Symmetry and Polar Duality of Polytopes
by David A. Cox
Symmetry 2015, 7(3), 1633-1645; https://doi.org/10.3390/sym7031633 - 10 Sep 2015
Cited by 6 | Viewed by 7078
Abstract
This expository article explores the connection between the polar duality from polyhedral geometry and mirror symmetry from mathematical physics and algebraic geometry. Topics discussed include duality of polytopes and cones as well as the famous quintic threefold and the toric variety of a [...] Read more.
This expository article explores the connection between the polar duality from polyhedral geometry and mirror symmetry from mathematical physics and algebraic geometry. Topics discussed include duality of polytopes and cones as well as the famous quintic threefold and the toric variety of a reflexive polytope. Full article
(This article belongs to the Special Issue Symmetry and Duality)
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