Symmetry

179 KiB

Open AccessArticle

Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries

by Renato Lemus

Symmetry 2012, 4(4), 667-685; https://doi.org/10.3390/sym4040667 - 30 Nov 2012

Cited by 18 | Viewed by 5897

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 [...] 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

► Show Figures

Figure 1

1042 KiB

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

by David A. Becker

Symmetry 2012, 4(4), 644-666; https://doi.org/10.3390/sym4040644 - 15 Nov 2012

Viewed by 11098

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 faces [...] 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)

► Show Figures

Figure 1

136 KiB

Open AccessArticle

Dirac Matrices and Feynman’s Rest of the Universe

by Young S. Kim and Marilyn E. Noz

Symmetry 2012, 4(4), 626-643; https://doi.org/10.3390/sym4040626 - 30 Oct 2012

Cited by 6 | Viewed by 5595

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 consists [...] 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

► Show Figures

Figure 1

321 KiB

Open AccessArticle

N = (4,4) Supersymmetry and T-Duality

by Malin Göteman

Symmetry 2012, 4(4), 603-625; https://doi.org/10.3390/sym4040603 - 24 Oct 2012

Cited by 2 | Viewed by 4745

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 only [...] 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

452 KiB

Open AccessArticle

Hexagonal Inflation Tilings and Planar Monotiles

by Michael Baake, Franz Gähler and Uwe Grimm

Symmetry 2012, 4(4), 581-602; https://doi.org/10.3390/sym4040581 - 22 Oct 2012

Cited by 11 | Viewed by 8557

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, which [...] 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)

► Show Figures

Figure 1

244 KiB

Open AccessArticle

On the Notions of Symmetry and Aperiodicity for Delone Sets

by Michael Baake and Uwe Grimm

Symmetry 2012, 4(4), 566-580; https://doi.org/10.3390/sym4040566 - 10 Oct 2012

Cited by 5 | Viewed by 7161

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 concept [...] 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)

► Show Figures

Figure 1

Journal Menu

Journal Browser

Symmetry, Volume 4, Issue 4 (December 2012) – 6 articles , Pages 566-685

Further Information

Guidelines

MDPI Initiatives

Follow MDPI