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Symmetry, Volume 4, Issue 4 (December 2012), Pages 566-685

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Research

Open AccessArticle On the Notions of Symmetry and Aperiodicity for Delone Sets
Symmetry 2012, 4(4), 566-580; doi:10.3390/sym4040566
Received: 3 August 2012 / Revised: 25 September 2012 / Accepted: 28 September 2012 / Published: 10 October 2012
Cited by 3 | PDF Full-text (244 KB) | HTML Full-text | XML Full-text
Abstract
Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the [...] Read more.
Non-periodic systems have become more important in recent years, both theoretically and practically. Their description via Delone sets requires the extension of many standard concepts of crystallography. Here, we summarise some useful notions of symmetry and aperiodicity, with special focus on the concept of the hull of a Delone set. Our aim is to contribute to a more systematic and consistent use of the different notions. Full article
(This article belongs to the Special Issue Crystal Symmetry and Structure)
Open AccessArticle Hexagonal Inflation Tilings and Planar Monotiles
Symmetry 2012, 4(4), 581-602; doi:10.3390/sym4040581
Received: 2 September 2012 / Revised: 8 October 2012 / Accepted: 14 October 2012 / Published: 22 October 2012
Cited by 5 | PDF Full-text (452 KB) | HTML Full-text | XML Full-text
Abstract
Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, [...] Read more.
Aperiodic tilings with a small number of prototiles are of particular interest, both theoretically and for applications in crystallography. In this direction, many people have tried to construct aperiodic tilings that are built from a single prototile with nearest neighbour matching rules, which is then called a monotile. One strand of the search for a planar monotile has focused on hexagonal analogues of Wang tiles. This led to two inflation tilings with interesting structural details. Both possess aperiodic local rules that define hulls with a model set structure. We review them in comparison, and clarify their relation with the classic half-hex tiling. In particular, we formulate various known results in a more comparative way, and augment them with some new results on the geometry and the topology of the underlying tiling spaces. Full article
(This article belongs to the Special Issue Polyhedra)
Open AccessArticle N = (4,4) Supersymmetry and T-Duality
Symmetry 2012, 4(4), 603-625; doi:10.3390/sym4040603
Received: 3 September 2012 / Revised: 12 October 2012 / Accepted: 15 October 2012 / Published: 24 October 2012
Cited by 1 | PDF Full-text (321 KB) | HTML Full-text | XML Full-text
Abstract
A sigma model with four-dimensional target space parametrized by chiral and twisted chiral N =(2,2) superfields can be extended to N =(4,4) supersymmetry off-shell, but this is not true for a model of semichiral fields, where the N = (4,4) supersymmetry can [...] Read more.
A sigma model with four-dimensional target space parametrized by chiral and twisted chiral N =(2,2) superfields can be extended to N =(4,4) supersymmetry off-shell, but this is not true for a model of semichiral fields, where the N = (4,4) supersymmetry can only be realized on-shell. The two models can be related to each other by T-duality. In this paper we perform a duality transformation from a chiral and twisted chiral model with off-shell N = (4,4) supersymmetry to a semichiral model. We find that additional non-linear terms must be added to the original transformations to obtain a semichiral model with N =(4,4) supersymmetry, and that the algebra closes on-shell as a direct consequence of the T-duality. Full article
Open AccessArticle Dirac Matrices and Feynman’s Rest of the Universe
Symmetry 2012, 4(4), 626-643; doi:10.3390/sym4040626
Received: 25 June 2012 / Revised: 6 October 2012 / Accepted: 23 October 2012 / Published: 30 October 2012
Cited by 1 | PDF Full-text (136 KB) | HTML Full-text | XML Full-text
Abstract
There are two sets of four-by-four matrices introduced by Dirac. The first set consists of fifteen Majorana matrices derivable from his four γ matrices. These fifteen matrices can also serve as the generators of the group SL(4, r). The second set [...] Read more.
There are two sets of four-by-four matrices introduced by Dirac. The first set consists of fifteen Majorana matrices derivable from his four γ matrices. These fifteen matrices can also serve as the generators of the group SL(4, r). The second set consists of ten generators of the Sp(4) group which Dirac derived from two coupled harmonic oscillators. It is shown possible to extend the symmetry of Sp(4) to that of SL(4, r) if the area of the phase space of one of the oscillators is allowed to become smaller without a lower limit. While there are no restrictions on the size of phase space in classical mechanics, Feynman’s rest of the universe makes this Sp(4)-to-SL(4, r) transition possible. The ten generators are for the world where quantum mechanics is valid. The remaining five generators belong to the rest of the universe. It is noted that the groups SL(4, r) and Sp(4) are locally isomorphic to the Lorentz groups O(3, 3) and O(3, 2) respectively. This allows us to interpret Feynman’s rest of the universe in terms of space-time symmetry. Full article
Open AccessArticle A Peculiarly Cerebroid Convex Zygo-Dodecahedron is an Axiomatically Balanced “House of Blues”: The Circle of Fifths to the Circle of Willis to Cadherin Cadenzas
Symmetry 2012, 4(4), 644-666; doi:10.3390/sym4040644
Received: 24 August 2012 / Revised: 28 October 2012 / Accepted: 6 November 2012 / Published: 15 November 2012
PDF Full-text (1042 KB) | HTML Full-text | XML Full-text
Abstract
A bilaterally symmetrical convex dodecahedron consisting of twelve quadrilateral faces is derived from the icosahedron via a process akin to Fuller’s Jitterbug Transformation. The unusual zygomorphic dodecahedron so obtained is shown to harbor a bilaterally symmetrical jazz/blues harmonic code on its twelve [...] Read more.
A bilaterally symmetrical convex dodecahedron consisting of twelve quadrilateral faces is derived from the icosahedron via a process akin to Fuller’s Jitterbug Transformation. The unusual zygomorphic dodecahedron so obtained is shown to harbor a bilaterally symmetrical jazz/blues harmonic code on its twelve faces that is related to such fundamental music theoretical constructs as the Circle of Fifths and Euler’s tonnetz. Curiously, the patterning within the aforementioned zygo-dodecahedron is discernibly similar to that observed in a ventral view of the human brain. Moreover, this same pattern is arguably evident during development of the embryonic pharynx. A possible role for the featured zygo-dodecahedron in cephalogenesis is considered. Recent studies concerning type II cadherins, an important class of proteins that promote cell adhesion, have generated data that is demonstrated to conform to this zygo-dodecahedral brain model in a substantially congruous manner. Full article
(This article belongs to the Special Issue Polyhedra)
Open AccessArticle Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries
Symmetry 2012, 4(4), 667-685; doi:10.3390/sym4040667
Received: 1 August 2012 / Revised: 23 November 2012 / Accepted: 26 November 2012 / Published: 30 November 2012
Cited by 6 | PDF Full-text (179 KB) | HTML Full-text | XML Full-text
Abstract
The eigenfunction approach used for discrete symmetries is deduced from the concept of quantum numbers. We show that the irreducible representations (irreps) associated with the eigenfunctions are indeed a shorthand notation for the set of eigenvalues of the class operators (character table). [...] Read more.
The eigenfunction approach used for discrete symmetries is deduced from the concept of quantum numbers. We show that the irreducible representations (irreps) associated with the eigenfunctions are indeed a shorthand notation for the set of eigenvalues of the class operators (character table). The need of a canonical chain of groups to establish a complete set of commuting operators is emphasized. This analysis allows us to establish in natural form the connection between the quantum numbers and the eigenfunction method proposed by J.Q. Chen to obtain symmetry adapted functions. We then proceed to present a friendly version of the eigenfunction method to project functions. Full article

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