Symmetry
http://www.mdpi.com/journal/symmetry
Latest open access articles published in Symmetry at http://www.mdpi.com/journal/symmetry<![CDATA[Symmetry, Vol. 6, Pages 210-221: Metalloporphyrin Symmetry in Chiral Recognition and Enantioselective Catalysis]]>
http://www.mdpi.com/2073-8994/6/2/210
Symmetry plays a fundamental role in chiral recognition and enantioselective catalysis. Porphyrins possess a number of structural features that make them attractive for the stereocontrol of chiral recognition and metal-catalyzed asymmetric reactions. This article is a brief account of our studies on chiral recognition and enantioselective catalysis by optically active metalloporphyrins. Some of the studies on chiral recognition and asymmetric catalysis by metalloporphyrins performed by others have also been included when useful.Symmetry2014-04-1062Review10.3390/sym60202102102212073-89942014-04-10doi: 10.3390/sym6020210Gérard SimonneauxHassan SrourPaul MauxSoizic ChevanceDaniel Carrie<![CDATA[Symmetry, Vol. 6, Pages 189-209: Topological Symmetry Groups of Small Complete Graphs]]>
http://www.mdpi.com/2073-8994/6/2/189
Topological symmetry groups were originally introduced to study the symmetries of non-rigid molecules, but have since been used to study the symmetries of any graph embedded in R3. In this paper, we determine for each complete graph Kn with n ≤ 6, what groups can occur as topological symmetry groups or orientation preserving topological symmetry groups of some embedding of the graph in R3.Symmetry2014-04-0862Article10.3390/sym60201891892092073-89942014-04-08doi: 10.3390/sym6020189Dwayne ChambersErica Flapan<![CDATA[Symmetry, Vol. 6, Pages 171-188: Development of Symmetry Concepts for Aperiodic Crystals]]>
http://www.mdpi.com/2073-8994/6/2/171
An overview is given of the use of symmetry considerations for aperiodic crystals. Superspace groups were introduced in the seventies for the description of incommensurate modulated phases with one modulation vector. Later, these groups were also used for quasi-periodic crystals of arbitrary rank. Further extensions use time reversal and time translation operations on magnetic and electrodynamic systems. An alternative description of magnetic structures to that with symmetry groups, the Shubnikov groups, is using representations of space groups. The same can be done for aperiodic crystals. A discussion of the relation between the two approaches is given. Representations of space groups and superspace groups play a role in the study of physical properties. These, and generalizations of them, are discussed for aperiodic crystals. They are used, in particular, for the characterization of phase transitions between aperiodic crystal phases.Symmetry2014-03-3162Article10.3390/sym60201711711882073-89942014-03-31doi: 10.3390/sym6020171Ted Janssen<![CDATA[Symmetry, Vol. 6, Pages 164-170: Spacetime Metrics from Gauge Potentials]]>
http://www.mdpi.com/2073-8994/6/2/164
I present an approach to gravity in which the spacetime metric is constructed from a non-Abelian gauge potential with values in the Lie algebra of the group U(2) (or the Lie algebra of quaternions). If the curvature of this potential vanishes, the metric reduces to a canonical curved background form reminiscent of the Friedmann S3 cosmological metric.Symmetry2014-03-2762Article10.3390/sym60201641641702073-89942014-03-27doi: 10.3390/sym6020164Ettore Minguzzi<![CDATA[Symmetry, Vol. 6, Pages 148-163: Symmetry Aspects of Dislocation-Effected Crystal Properties: Material Strength Levels and X-ray Topographic Imaging]]>
http://www.mdpi.com/2073-8994/6/1/148
Several materials science type research topics are described in which advantageous use of crystal symmetry considerations has been helpful in ferreting the essential elements of dislocation behavior in determining material properties or for characterizing crystal/polycrystalline structural relationships; for example: (1) the mechanical strengthening produced by a symmetrical bicrystal grain boundary; (2) cleavage crack formation at the intersection within a crystal of symmetrical dislocation pile-ups; (3) symmetry aspects of anisotropic crystal indentation hardness measurements; (4) X-ray diffraction topography imaging of dislocation strains and subgrain boundary misorientations; and (5) point and space group aspects of twinning. Several applications are described in relation to the strengthening of grain boundaries in nanopolycrystals and of multiply-oriented crystal grains in polysilicon photovoltaic solar cell materials. A number of crystallographic aspects of the different topics are illustrated with a stereographic method of presentation.Symmetry2014-03-2061Review10.3390/sym60101481481632073-89942014-03-20doi: 10.3390/sym6010148Ronald Armstrong<![CDATA[Symmetry, Vol. 6, Pages 111-147: The Hunt for Supersymmetry at the Tevatron]]>
http://www.mdpi.com/2073-8994/6/1/111
During the Tevatron data-taking period from April 2001 to September 2011 (Run-II), several searches for supersymmetric particles were performed. The results from searches by the CDF and DØ collaborations are concisely reviewed. This includes results up to the summer conferences of 2013. Model-independent and model-dependent limits on new particle production are set, and interpretations in supersymmetric models are given. Several limits from the Large Electron Positron (LEP) era have been extended. Specific results are placed into the context of the Tevatron performance expectations and a few of the current results from searches at the Large Hadron Collider (LHC).Symmetry2014-03-1961Review10.3390/sym60101111111472073-89942014-03-19doi: 10.3390/sym6010111André Sopczak<![CDATA[Symmetry, Vol. 6, Pages 103-110: Pseudo Hermitian Interactions in the Dirac Equation]]>
http://www.mdpi.com/2073-8994/6/1/103
We consider a (2 + 1)-dimensional massless Dirac equation in the presence of complex vector potentials. It is shown that such vector potentials (leading to complex magnetic fields) can produce bound states, and the Dirac Hamiltonians are η-pseudo Hermitian. Some examples have been explicitly worked out.Symmetry2014-03-1761Article10.3390/sym60101031031102073-89942014-03-17doi: 10.3390/sym6010103Orlando PanellaPinaki Roy<![CDATA[Symmetry, Vol. 6, Pages 90-102: Using Symmetries (Beyond Geometric Symmetries) in Chemical Computations: Computing Parameters of Multiple Binding Sites]]>
http://www.mdpi.com/2073-8994/6/1/90
We show how transformation group ideas can be naturally used to generate efficient algorithms for scientific computations. The general approach is illustrated on the example of determining, from the experimental data, the dissociation constants related to multiple binding sites. We also explain how the general transformation group approach is related to the standard (backpropagation) neural networks; this relation justifies the potential universal applicability of the group-related approach.Symmetry2014-02-2561Article10.3390/sym6010090901022073-89942014-02-25doi: 10.3390/sym6010090Andres OrtizVladik Kreinovich<![CDATA[Symmetry, Vol. 6, Pages 89: Acknowledgement to Reviewers of Symmetry in 2013]]>
http://www.mdpi.com/2073-8994/6/1/89
The editors of Symmetry would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2013.Symmetry2014-02-2461Editorial10.3390/sym601008989892073-89942014-02-24doi: 10.3390/sym6010089 Symmetry Editorial Office<![CDATA[Symmetry, Vol. 6, Pages 67-88: On General Off-Shell Representations of World Line (1D) Supersymmetry]]>
http://www.mdpi.com/2073-8994/6/1/67
Every finite-dimensional unitary representation of the N-extended world line supersymmetry without central charges may be obtained by a sequence of differential transformations from a direct sum of minimal Adinkras, simple supermultiplets that are identifiable with representations of the Clifford algebra. The data specifying this procedure is a sequence of subspaces of the direct sum of Adinkras, which then opens an avenue for the classification of the continuum of the so-constructed off-shell supermultiplets.Symmetry2014-02-0361Article10.3390/sym601006767882073-89942014-02-03doi: 10.3390/sym6010067Charles DoranTristan HübschKevin IgaGregory Landweber<![CDATA[Symmetry, Vol. 6, Pages 23-66: Symmetry-Breaking in a Rate Model for a Biped Locomotion Central Pattern Generator]]>
http://www.mdpi.com/2073-8994/6/1/23
The timing patterns of animal gaits are produced by a network of spinal neurons called a Central Pattern Generator (CPG). Pinto and Golubitsky studied a four-node CPG for biped dynamics in which each leg is associated with one flexor node and one extensor node, with Ζ2 x Ζ2 symmetry. They used symmetric bifurcation theory to predict the existence of four primary gaits and seven secondary gaits. We use methods from symmetric bifurcation theory to investigate local bifurcation, both steady-state and Hopf, for their network architecture in a rate model. Rate models incorporate parameters corresponding to the strengths of connections in the CPG: positive for excitatory connections and negative for inhibitory ones. The three-dimensional space of connection strengths is partitioned into regions that correspond to the first local bifurcation from a fully symmetric equilibrium. The partition is polyhedral, and its symmetry group is that of a tetrahedron. It comprises two concentric tetrahedra, subdivided by various symmetry planes. The tetrahedral symmetry arises from the structure of the eigenvalues of the connection matrix, which is involved in, but not equal to, the Jacobian of the rate model at bifurcation points. Some of the results apply to rate equations on more general networks.Symmetry2014-01-0361Article10.3390/sym601002323662073-89942014-01-03doi: 10.3390/sym6010023Ian Stewart<![CDATA[Symmetry, Vol. 6, Pages 1-22: Peripheral Contour Grouping and Saccade Targeting: The Role of Mirror Symmetry]]>
http://www.mdpi.com/2073-8994/6/1/1
Integrating shape contours in the visual periphery is vital to our ability to locate objects and thus make targeted saccadic eye movements to efficiently explore our surroundings. We tested whether global shape symmetry facilitates peripheral contour integration and saccade targeting in three experiments, in which observers responded to a successful peripheral contour detection by making a saccade towards the target shape. The target contours were horizontally (Experiment 1) or vertically (Experiments 2 and 3) mirror symmetric. Observers responded by making a horizontal (Experiments 1 and 2) or vertical (Experiment 3) eye movement. Based on an analysis of the saccadic latency and accuracy, we conclude that the figure-ground cue of global mirror symmetry in the periphery has little effect on contour integration or on the speed and precision with which saccades are targeted towards objects. The role of mirror symmetry may be more apparent under natural viewing conditions with multiple objects competing for attention, where symmetric regions in the visual field can pre-attentively signal the presence of objects, and thus attract eye movements.Symmetry2014-01-0261Article10.3390/sym60100011222073-89942014-01-02doi: 10.3390/sym6010001Michaël SassiMaarten DemeyerJohan Wagemans<![CDATA[Symmetry, Vol. 5, Pages 344-354: Effect of Symmetry Breaking on Electronic Band Structure: Gap Opening at the High Symmetry Points]]>
http://www.mdpi.com/2073-8994/5/4/344
Some characteristic features of band structures, like the band degeneracy at high symmetry points or the existence of energy gaps, usually reflect the symmetry of the crystal or, more precisely, the symmetry of the wave vector group at the relevant points of the Brillouin zone. In this paper, we will illustrate this property by considering two-dimensional (2D)-hexagonal lattices characterized by a possible two-fold degenerate band at the K points with a linear dispersion (Dirac points). By combining scanning tunneling spectroscopy and angle-resolved photoemission, we study the electronic properties of a similar system: the Ag/Cu(111) interface reconstruction characterized by a hexagonal superlattice, and we show that the gap opening at the K points of the Brillouin zone of the reconstructed cell is due to the symmetry breaking of the wave vector group.Symmetry2013-12-0954Article10.3390/sym50403443443542073-89942013-12-09doi: 10.3390/sym5040344Guillaume VasseurYannick Fagot-RevuratBertrand KierrenMuriel SicotDaniel Malterre<![CDATA[Symmetry, Vol. 5, Pages 313-343: Interplay between Point-Group Symmetries and the Choice of the Bloch Basis in Multiband Models]]>
http://www.mdpi.com/2073-8994/5/4/313
We analyze the point-group symmetries of generic multiband tight-binding models with respect to the transformation properties of the effective interactions. While the vertex functions in the orbital language may transform non-trivially under point-group operations, their point-group behavior in the band language can be simplified by choosing a suitable Bloch basis. We first give two analytically accessible examples. Then, we show that, for a large class of models, a natural Bloch basis exists, in which the vertex functions in the band language transform trivially under all point-group operations. As a consequence, the point-group symmetries can be used to reduce the computational effort in perturbative many-particle approaches, such as the functional renormalization group.Symmetry2013-11-1154Article10.3390/sym50403133133432073-89942013-11-11doi: 10.3390/sym5040313Stefan MaierCarsten HonerkampQiang-Hua Wang<![CDATA[Symmetry, Vol. 5, Pages 287-312: Multiple Solutions to Implicit Symmetric Boundary Value Problems for Second Order Ordinary Differential Equations (ODEs): Equivariant Degree Approach]]>
http://www.mdpi.com/2073-8994/5/4/287
In this paper, we develop a general framework for studying Dirichlet Boundary Value Problems (BVP) for second order symmetric implicit differential systems satisfying the Hartman-Nagumo conditions, as well as a certain non-expandability condition. The main result, obtained by means of the equivariant degree theory, establishes the existence of multiple solutions together with a complete description of their symmetric properties. The abstract result is supported by a concrete example of an implicit system respecting D4-symmetries.Symmetry2013-11-0754Article10.3390/sym50402872873122073-89942013-11-07doi: 10.3390/sym5040287Zalman BalanovWieslaw KrawcewiczZhichao LiMylinh Nguyen<![CDATA[Symmetry, Vol. 5, Pages 271-286: A New Route to the Majorana Equation]]>
http://www.mdpi.com/2073-8994/5/4/271
In this paper, we suggest an alternative strategy to derive the complex two-component Majorana equation with a mass term and elucidate the related Lorentz transformation. The Majorana equation is established completely on its own, rather than derived from the chiral Dirac equation. Thereby, use is made of the complex conjugation operator and Pauli spin matrices only. The eigenfunctions of the two-component complex Majorana equation are also calculated. The associated quantum fields are found to describe particles and antiparticles, which have opposite mean helicities and are not their own antiparticles, but correspond to two independent degrees of freedom. The four-component real Dirac equation in its Majorana representation is shown to be the natural outcome of the two-component complex Majorana equation. Both types of equations come in two forms, which correspond to the irreducible left- and right-chiral representations of the Lorentz group.Symmetry2013-09-2654Article10.3390/sym50402712712862073-89942013-09-26doi: 10.3390/sym5040271Eckart Marsch<![CDATA[Symmetry, Vol. 5, Pages 253-270: Supersymmetric Version of the Euler System and Its Invariant Solutions]]>
http://www.mdpi.com/2073-8994/5/3/253
In this paper, we formulate a supersymmetric extension of the Euler system of equations. We compute a superalgebra of Lie symmetries of the supersymmetric system. Next, we classify the one-dimensional subalgebras of this superalgebra into 49 equivalence conjugation classes. For some of the subalgebras, the invariants have a non-standard structure. For nine selected subalgebras, we use the symmetry reduction method to find invariants, orbits and reduced systems. Through the solutions of these reduced systems, we obtain solutions of the supersymmetric Euler system. The obtained solutions include bumps, kinks, multiple wave solutions and solutions expressed in terms of arbitrary functions.Symmetry2013-07-1253Article10.3390/sym50302532532702073-89942013-07-12doi: 10.3390/sym5030253A. GrundlandAlexander Hariton<![CDATA[Symmetry, Vol. 5, Pages 233-252: Symmetries Shared by the Poincaré Group and the Poincaré Sphere]]>
http://www.mdpi.com/2073-8994/5/3/233
Henri Poincaré formulated the mathematics of Lorentz transformations, known as the Poincaré group. He also formulated the Poincaré sphere for polarization optics. It is shown that these two mathematical instruments can be derived from the two-by-two representations of the Lorentz group. Wigner’s little groups for internal space-time symmetries are studied in detail. While the particle mass is a Lorentz-invariant quantity, it is shown to be possible to address its variations in terms of the decoherence mechanism in polarization optics.Symmetry2013-06-2753Article10.3390/sym50302332332522073-89942013-06-27doi: 10.3390/sym5030233Young KimMarilyn Noz<![CDATA[Symmetry, Vol. 5, Pages 215-232: Fermi Surface Reconstruction due to Hidden Rotating Antiferromagnetism in N and P-Type High-TC Cuprates]]>
http://www.mdpi.com/2073-8994/5/2/215
The Fermi surface calculated within the rotating antiferromagnetism theory undergoes a topological change when doping changes from p-type to n-type, in qualitative agreement with experimental data for n-type cuprate Nd2−xCexCuO4 and p-type La2−xSrxCuO4. Also, the reconstruction of the Fermi surface, observed experimentally close to optimal doping in p-type cuprates, and slightly higher than optimal doping in the overdoped regime for this n-type high-TC cuprate, is well accounted for in this theory. This reconstruction is a consequence of the quantum criticality caused by the disappearance of rotating antiferromagnetism. The present results are in qualitative agreement with recently observed quantum oscillations in some high-TC cuprates. This paper presents new results about the application of the rotating antiferromagnetism theory to the study of the electronic structure for n-type materials.Symmetry2013-05-0752Article10.3390/sym50202152152322073-89942013-05-07doi: 10.3390/sym5020215Mohamed Azzouz<![CDATA[Symmetry, Vol. 5, Pages 119-214: Topological Many-Body States in Quantum Antiferromagnets via Fuzzy Supergeometry]]>
http://www.mdpi.com/2073-8994/5/2/119
Recent vigorous investigations of topological order have not only discovered new topological states of matter, but also shed new light on “already known” topological states. One established example with topological order is the valence bond solid (VBS) states in quantum antiferromagnets. The VBS states are disordered spin liquids with no spontaneous symmetry breaking, but most typically manifest a topological order known as a hidden string order on the 1D chain. Interestingly, the VBS models are based on mathematics analogous to fuzzy geometry. We review applications of the mathematics of fuzzy supergeometry in the construction of supersymmetric versions of VBS (SVBS) states and give a pedagogical introduction of SVBS models and their properties. As concrete examples, we present detailed analysis of supersymmetric versions of SU(2) and SO(5) VBS states, i.e., UOSp(N|2) and UOSp(N|4) SVBS states, whose mathematics are closely related to fuzzy two- and four-superspheres. The SVBS states are physically interpreted as hole-doped VBS states with a superconducting property that interpolates various VBS states, depending on the value of a hole-doping parameter. The parent Hamiltonians for SVBS states are explicitly constructed, and their gapped excitations are derived within the single-mode approximation on 1D SVBS chains. Prominent features of the SVBS chains are discussed in detail, such as a generalized string order parameter and entanglement spectra. It is realized that the entanglement spectra are at least doubly degenerate, regardless of the parity of bulk (super)spins. The stability of the topological phase with supersymmetry is discussed, with emphasis on its relation to particular edge (super)spin states.Symmetry2013-04-2652Review10.3390/sym50201191192142073-89942013-04-26doi: 10.3390/sym5020119Kazuki HasebeKeisuke Totsuka<![CDATA[Symmetry, Vol. 5, Pages 86-118: Frame Transformation Relations and Symmetry Analysis of Fluxional Symmetric Rotor Dimers]]>
http://www.mdpi.com/2073-8994/5/1/86
The theory of Frame transformation relations between the states of Born Oppenheimer and the weak coupling approximations is developed for polyatomic molecules. The symmetry relations are a generalization of the frame transformation relations derived by Harter and Crogman for coupled rotor molecules. A key internal symmetry label (named “soul”) is defined so that it remains a constant label for frame transformation relations, and is conserved during vibronic transitions, ionization, and even dissociation provided the nuclear spin-rotation interaction is relatively small. Simplified procedures are given for obtaining selection rules, statistical weights, and matrix elements of multipole operators for common molecules having various point symmetries.Symmetry2013-02-0551Article10.3390/sym5010086861182073-89942013-02-05doi: 10.3390/sym5010086Horace CrogmanBumgyunmiga ChoiHarrison ChenWilliam Harter<![CDATA[Symmetry, Vol. 5, Pages 81-85: Perlman and Wellner’s Circular and Transformed Circular Copulas are Particular Beta and t Copulas]]>
http://www.mdpi.com/2073-8994/5/1/81
All but one of the copulas in a recent paper in Symmetry by Perlman and Wellner can be identified as particular members of either the beta or t families of elliptical copulas.Symmetry2013-01-3151Short Note10.3390/sym501008181852073-89942013-01-31doi: 10.3390/sym5010081M. C. Jones<![CDATA[Symmetry, Vol. 5, Pages 54-80: Non-Crystallographic Symmetry in Packing Spaces]]>
http://www.mdpi.com/2073-8994/5/1/54
In the following, isomorphism of an arbitrary finite group of symmetry, non-crystallographic symmetry (quaternion groups, Pauli matrices groups, and other abstract subgroups), in addition to the permutation group, are considered. Application of finite groups of permutations to the packing space determines space tilings by policubes (polyominoes) and forms a structure. Such an approach establishes the computer design of abstract groups of symmetry. Every finite discrete model of the real structure is an element of symmetry groups, including non-crystallographic ones. The set packing spaces of the same order N characterizes discrete deformation transformations of the structure.Symmetry2013-01-0951Article10.3390/sym501005454802073-89942013-01-09doi: 10.3390/sym5010054Valery RauLeonty LomtevTamara Rau<![CDATA[Symmetry, Vol. 5, Pages 47-53: A Note on Lower Bounds for Colourful Simplicial Depth]]>
http://www.mdpi.com/2073-8994/5/1/47
The colourful simplicial depth problem in dimension d is to find a configuration of (d+1) sets of (d+1) points such that the origin is contained in the convex hull of each set, or colour, but contained in a minimal number of colourful simplices generated by taking one point from each set. A construction attaining d2 + 1 simplices is known, and is conjectured to be minimal. This has been confirmed up to d = 3, however the best known lower bound for d ≥ 4 is ⌈(d+1)2 /2 ⌉. In this note, we use a branching strategy to improve the lower bound in dimension 4 from 13 to 14.Symmetry2013-01-0751Short Note10.3390/sym501004747532073-89942013-01-07doi: 10.3390/sym5010047Antoine DezaTamon StephenFeng Xie<![CDATA[Symmetry, Vol. 5, Pages 1-46: Taylor–Socolar Hexagonal Tilings as Model Sets]]>
http://www.mdpi.com/2073-8994/5/1/1
The Taylor–Socolar tilings are regular hexagonal tilings of the plane but are distinguished in being comprised of hexagons of two colors in an aperiodic way. We place the Taylor–Socolar tilings into an algebraic setting, which allows one to see them directly as model sets and to understand the corresponding tiling hull along with its generic and singular parts. Although the tilings were originally obtained by matching rules and by substitution, our approach sets the tilings into the framework of a cut and project scheme and studies how the tilings relate to the corresponding internal space. The centers of the entire set of tiles of one tiling form a lattice Q in the plane. If XQ denotes the set of all Taylor–Socolar tilings with centers on Q, then XQ forms a natural hull under the standard local topology of hulls and is a dynamical system for the action of Q.The Q-adic completion Q of Q is a natural factor of XQ and the natural mapping XQ → Q is bijective except at a dense set of points of measure 0 in /Q. We show that XQ consists of three LI classes under translation. Two of these LI classes are very small, namely countable Q-orbits in XQ. The other is a minimal dynamical system, which maps surjectively to /Q and which is variously 2 : 1, 6 : 1, and 12 : 1 at the singular points. We further develop the formula of what determines the parity of the tiles of a tiling in terms of the coordinates of its tile centers. Finally we show that the hull of the parity tilings can be identified with the hull XQ; more precisely the two hulls are mutually locally derivable.Symmetry2012-12-2851Article10.3390/sym50100011462073-89942012-12-28doi: 10.3390/sym5010001Jeong-Yup LeeRobert Moody<![CDATA[Symmetry, Vol. 4, Pages 667-685: Quantum Numbers and the Eigenfunction Approach to Obtain Symmetry Adapted Functions for Discrete Symmetries]]>
http://www.mdpi.com/2073-8994/4/4/667
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.Symmetry2012-11-3044Article10.3390/sym40406676676852073-89942012-11-30doi: 10.3390/sym4040667Renato Lemus<![CDATA[Symmetry, Vol. 4, Pages 644-666: 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]]>
http://www.mdpi.com/2073-8994/4/4/644
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.Symmetry2012-11-1544Article10.3390/sym40406446446662073-89942012-11-15doi: 10.3390/sym4040644David Becker<![CDATA[Symmetry, Vol. 4, Pages 626-643: Dirac Matrices and Feynman’s Rest of the Universe]]>
http://www.mdpi.com/2073-8994/4/4/626
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.Symmetry2012-10-3044Article10.3390/sym40406266266432073-89942012-10-30doi: 10.3390/sym4040626Young S. KimMarilyn E. Noz<![CDATA[Symmetry, Vol. 4, Pages 603-625: N = (4,4) Supersymmetry and T-Duality]]>
http://www.mdpi.com/2073-8994/4/4/603
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.Symmetry2012-10-2444Article10.3390/sym40406036036252073-89942012-10-24doi: 10.3390/sym4040603Malin Göteman<![CDATA[Symmetry, Vol. 4, Pages 581-602: Hexagonal Inflation Tilings and Planar Monotiles]]>
http://www.mdpi.com/2073-8994/4/4/581
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.Symmetry2012-10-2244Article10.3390/sym40405815816022073-89942012-10-22doi: 10.3390/sym4040581Michael BaakeFranz GählerUwe Grimm<![CDATA[Symmetry, Vol. 4, Pages 566-580: On the Notions of Symmetry and Aperiodicity for Delone Sets]]>
http://www.mdpi.com/2073-8994/4/4/566
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.Symmetry2012-10-1044Article10.3390/sym40405665665802073-89942012-10-10doi: 10.3390/sym4040566Michael BaakeUwe Grimm<![CDATA[Symmetry, Vol. 4, Pages 545-565: Barrel Pseudotilings]]>
http://www.mdpi.com/2073-8994/4/3/545
This paper describes 4-valent tiling-like structures, called pseudotilings, composed of barrel tiles and apeirogonal pseudotiles in Euclidean 3-space. These (frequently face-to-face) pseudotilings naturally rise in columns above 3-valent plane tilings by convex polygons, such that each column is occupied by stacked congruent barrel tiles or congruent apeirogonal pseudotiles. No physical space is occupied by the apeirogonal pseudotiles. Many interesting pseudotilings arise from plane tilings with high symmetry. As combinatorial structures, these are abstract polytopes of rank 4 with both finite and infinite 2-faces and facets.Symmetry2012-08-3043Article10.3390/sym40305455455652073-89942012-08-30doi: 10.3390/sym4030545Undine Leopold<![CDATA[Symmetry, Vol. 4, Pages 537-544: A Higher Dimensional Description of the Structure of β-Mn]]>
http://www.mdpi.com/2073-8994/4/3/537
The structure of β-Mn crystallizes in space group P4132. The pseudo 8-fold nature of the 41 axes makes it constitute an approximant to the octagonal quasicrystals. In this paper we analyze why a five-dimensional super space group containing mutually perpendicular 8-fold axes cannot generate P4132 on projection to 3-d space and how this may instead be accomplished from a six-dimensional model. A procedure for generating the actual structure of β-Mn lifted to six-dimensional space is given.Symmetry2012-08-2743Article10.3390/sym40305375375442073-89942012-08-27doi: 10.3390/sym4030537Sven LidinDaniel Fredrickson<![CDATA[Symmetry, Vol. 4, Pages 517-536: Supersymmetric Extensions of Non-Relativistic Scaling Algebras]]>
http://www.mdpi.com/2073-8994/4/3/517
An exciting subject in string theory is to consider some applications of the AdS/CFT correspondence to realistic systems like condensed matter systems. Since most of such systems are non-relativistic, an anisotropic scaling symmetry with the general value of dynamical critical exponent z plays an important role in constructing the gravity duals for non-relativistic field theories. Supersymmetric extensions of symmetry algebras including the anisotropic scaling are very helpful to consider holographic relations accurately. We give a short summary on the classification of superalgebras with the anisotropic scaling as subalgebras of the following Lie superalgebras, psu(2,2|4), osp(8|4) and osp (8*|4), which appear in the study of AdS/CFT in type IIB string and M theories. It contains supersymmetric extensions of Schrödinger algebra and Lifshitz algebra.Symmetry2012-08-2443Review10.3390/sym40305175175362073-89942012-08-24doi: 10.3390/sym4030517Makoto SakaguchiKentaroh Yoshida<![CDATA[Symmetry, Vol. 4, Pages 507-516: Flexibility of Hydrogen Bond and Lowering of Symmetry in Proton Conductor]]>
http://www.mdpi.com/2073-8994/4/3/507
In order to investigate why crystal symmetry lowers with increasing temperature by phase transition of TII–III (=369 K) in Cs3H(SeO4)2, in spite of the fact that crystal symmetry in the high-temperature phase of many ionic conductors becomes higher by the phase transition, we have studied the relation between the change in crystal symmetry and the appearance of proton motion. It was found from the analysis of domains based on crystal structure that the number of possible geometrical arrangement of hydrogen bond in phase II becomes two times larger than that in phase III, derived from the lowering of crystal symmetry with increasing temperature. These results indicate that the lowering of crystal symmetry in phase II appears by the increase of the number of geometrical arrangements and by the enhancement of the flexibility of hydrogen bond. Considering that the enhancement of the flexibility of hydrogen bond yields mobile proton in phase II, it is deduced that mobile proton in phase II appears in exchange for the lowering of crystal symmetry at II–III phase transition.Symmetry2012-08-2343Article10.3390/sym40305075075162073-89942012-08-23doi: 10.3390/sym4030507Yukihiko YoshidaJunko HatoriHinako KawakamiYasumitsu MatsuoSeiichiro Ikehata<![CDATA[Symmetry, Vol. 4, Pages 474-506: Supersymmetric Sigma Model Geometry]]>
http://www.mdpi.com/2073-8994/4/3/474
This is a review of how sigma models formulated in Superspace have become important tools for understanding geometry. Topics included are: The (hyper)kähler reduction; projective superspace; the generalized Legendre construction; generalized Kähler geometry and constructions of hyperkähler metrics on Hermitian symmetric spaces.Symmetry2012-08-2343Article10.3390/sym40304744745062073-89942012-08-23doi: 10.3390/sym4030474Ulf Lindström<![CDATA[Symmetry, Vol. 4, Pages 452-473: Supersymmetric Quantum Mechanics and Solvable Models]]>
http://www.mdpi.com/2073-8994/4/3/452
We review solvable models within the framework of supersymmetric quantum mechanics (SUSYQM). In SUSYQM, the shape invariance condition insures solvability of quantum mechanical problems. We review shape invariance and its connection to a consequent potential algebra. The additive shape invariance condition is specified by a difference-differential equation; we show that this equation is equivalent to an infinite set of partial differential equations. Solving these equations, we show that the known list of ħ-independent superpotentials is complete. We then describe how these equations could be extended to include superpotentials that do depend on ħ.Symmetry2012-08-1643Review10.3390/sym40304524524732073-89942012-08-16doi: 10.3390/sym4030452Jonathan BougieAsim GangopadhyayaJeffry MallowConstantin Rasinariu<![CDATA[Symmetry, Vol. 4, Pages 441-451: Soliton and Similarity Solutions of Ν = 2, 4 Supersymmetric Equations]]>
http://www.mdpi.com/2073-8994/4/3/441
We produce soliton and similarity solutions of supersymmetric extensions of Burgers, Korteweg–de Vries and modified KdV equations. We give new representations of the τ -functions in Hirota bilinear formalism. Chiral superfields are used to obtain such solutions. We also introduce new solitons called virtual solitons whose nonlinear interactions produce no phase shifts.Symmetry2012-08-0843Article10.3390/sym40304414414512073-89942012-08-08doi: 10.3390/sym4030441Laurent DelisleVéronique Hussin<![CDATA[Symmetry, Vol. 4, Pages 427-440: Duffin–Kemmer–Petiau and Dirac Equations—A Supersymmetric Connection]]>
http://www.mdpi.com/2073-8994/4/3/427
In the present paper we study subsolutions of the Dirac and Duffin–Kemmer–Petiau equations in the interacting case. It is shown that the Dirac equation in longitudinal external fields can be split into two covariant subequations (Dirac equations with built-in projection operators). Moreover, it is demonstrated that the Duffin–Kemmer–Petiau equations in crossed fields can be split into two 3 x 3 subequations. We show that all the subequations can be obtained via minimal coupling from the same 3 x 3 subequations which are thus a supersymmetric link between fermionic and bosonicdegrees of freedom.Symmetry2012-08-0743Article10.3390/sym40304274274402073-89942012-08-07doi: 10.3390/sym4030427Andrzej Okniński<![CDATA[Symmetry, Vol. 4, Pages 379-426: Symmetry-Adapted Fourier Series for the Wallpaper Groups]]>
http://www.mdpi.com/2073-8994/4/3/379
Two-dimensional (2D) functions with wallpaper group symmetry can be written as Fourier series displaying both translational and point-group symmetry. We elaborate the symmetry-adapted Fourier series for each of the 17 wallpaper groups. The symmetry manifests itself through constraints on and relations between the Fourier coefficients. Visualising the equivalencies of Fourier coefficients by means of discrete 2D maps reveals how direct-space symmetry is transformed into coefficient-space symmetry. Explicit expressions are given for the Fourier series and Fourier coefficient maps of both real and complex functions, readily applicable to the description of the properties of 2D materials like graphene or boron-nitride.Symmetry2012-07-1743Article10.3390/sym40303793794262073-89942012-07-17doi: 10.3390/sym4030379Bart Verberck<![CDATA[Symmetry, Vol. 4, Pages 344-378: Particle-Dependent Deformations of Lorentz Symmetry]]>
http://www.mdpi.com/2073-8994/4/3/344
I report results suggesting that it is possible to introduce laws of relativistic kinematics endowing different types of particles with suitably different deformed-Lorentz-symmetry properties. I also consider some possible applications of these results, among which I highlight those relevant for addressing a long-standing challenge in the description of composite particles, such as atoms, within quantum-gravity-inspired scenarios with Planck-scale deformations of Lorentz symmetry. Some of the new elements here introduced in the formulation of relativistic kinematics appear to also provide the starting point for the development of a correspondingly novel mathematical formulation of spacetime-symmetry algebras.Symmetry2012-07-0343Article10.3390/sym40303443443782073-89942012-07-03doi: 10.3390/sym4030344Giovanni Amelino-Camelia<![CDATA[Symmetry, Vol. 4, Pages 336-343: Superspheres: Intermediate Shapes between Spheres and Polyhedra]]>
http://www.mdpi.com/2073-8994/4/3/336
Using an x-y-z coordinate system, the equations of the superspheres have been extended to describe intermediate shapes between a sphere and various convex polyhedra. Near-polyhedral shapes composed of {100}, {111} and {110} surfaces with round edges are treated in the present study, where {100}, {111} and {110} are the Miller indices of crystals with cubic structures. The three parameters p, a and b are included to describe the {100}-{111}-{110} near-polyhedral shapes, where p describes the degree to which the shape is a polyhedron and a and b determine the ratios of the {100}, {111} and {110} surfaces.Symmetry2012-07-0343Article10.3390/sym40303363363432073-89942012-07-03doi: 10.3390/sym4030336Susumu Onaka<![CDATA[Symmetry, Vol. 4, Pages 329-335: Topological Invariance under Line Graph Transformations]]>
http://www.mdpi.com/2073-8994/4/2/329
It is shown that the line graph transformation G ↦ L(G) of a graph G preserves an isomorphic copy of G as the nerve of a finite simplicial complex K which is naturally associated with the Krausz decomposition of L(G). As a consequence, the homology of K is isomorphic to that of G. This homology invariance algebraically confirms several well known graph theoretic properties of line graphs and formally establishes the Euler characteristic of G as a line graph transformation invariant.Symmetry2012-06-0842Article10.3390/sym40203293293352073-89942012-06-08doi: 10.3390/sym4020329Allen D. Parks<![CDATA[Symmetry, Vol. 4, Pages 302-328: Knots in Art]]>
http://www.mdpi.com/2073-8994/4/2/302
We analyze applications of knots and links in the Ancient art, beginning from Babylonian, Egyptian, Greek, Byzantine and Celtic art. Construction methods used in art are analyzed on the examples of Celtic art and ethnical art of Tchokwe people from Angola or Tamil art, where knots are constructed as mirror-curves. We propose different methods for generating knots and links based on geometric polyhedra, suitable for applications in architecture and sculpture.Symmetry2012-06-0542Article10.3390/sym40203023023282073-89942012-06-05doi: 10.3390/sym4020302Slavik JablanLjiljana RadovićRadmila SazdanovićAna Zeković<![CDATA[Symmetry, Vol. 4, Pages 285-301: Diagrammatics in Art and Mathematics]]>
http://www.mdpi.com/2073-8994/4/2/285
This paper explores two-way relations between visualizations in mathematics and mathematical art, as well as art in general. A collection of vignettes illustrates connection points, including visualizing higher dimensions, tessellations, knots and links, plotting zeros of polynomials, and new and rapidly developing mathematical discipline, diagrammatic categorification.Symmetry2012-05-2242Article10.3390/sym40202852853012073-89942012-05-22doi: 10.3390/sym4020285Radmila Sazdanovic<![CDATA[Symmetry, Vol. 4, Pages 276-284: Following Knots down Their Energy Gradients]]>
http://www.mdpi.com/2073-8994/4/2/276
This paper details a series of experiments in searching for minimal energy configurations for knots and links using the computer program KnotPlot.Symmetry2012-04-2742Article10.3390/sym40202762762842073-89942012-04-27doi: 10.3390/sym4020276Louis H. Kauffman<![CDATA[Symmetry, Vol. 4, Pages 265-275: Maniplexes: Part 1: Maps, Polytopes, Symmetry and Operators]]>
http://www.mdpi.com/2073-8994/4/2/265
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.Symmetry2012-04-1642Article10.3390/sym40202652652752073-89942012-04-16doi: 10.3390/sym4020265Steve Wilson<![CDATA[Symmetry, Vol. 4, Pages 251-264: One-Sign Order Parameter in Iron Based Superconductor]]>
http://www.mdpi.com/2073-8994/4/1/251
The onset of superconductivity at the transition temperature is marked by the onset of order, which is characterized by an energy gap. Most models of the iron-based superconductors find a sign-changing (s±) order parameter [1–6], with the physical implication that pairing is driven by spin fluctuations. Recent work, however, has indicated that LiFeAs has a simple isotropic order parameter [7–9] and spin fluctuations are not necessary [7,10], contrary to the models [1–6]. The strength of the spin fluctuations has been controversial [11,12], meaning that the mechanism of superconductivity cannot as yet be determined. We report the momentum dependence of the superconducting energy gap, where we find an anisotropy that rules out coupling through spin fluctuations and the sign change. The results instead suggest that orbital fluctuations assisted by phonons [13,14] are the best explanation for superconductivity.Symmetry2012-03-2141Article10.3390/sym40102512512642073-89942012-03-21doi: 10.3390/sym4010251Sergey V. BorisenkoVolodymyr B. ZabolotnyyAlexnader A. KordyukDanil V. EvtushinskyTimur K. KimIgor V. MorozovRolf FollathBernd Büchner<![CDATA[Symmetry, Vol. 4, Pages 225-250: Classical Knot Theory]]>
http://www.mdpi.com/2073-8994/4/1/225
This paper is a very brief introduction to knot theory. It describes knot coloring by quandles, the fundamental group of a knot complement, and handle-decompositions of knot complements.Symmetry2012-03-0741Article10.3390/sym40102252252502073-89942012-03-07doi: 10.3390/sym4010225J. Scott Carter<![CDATA[Symmetry, Vol. 4, Pages 219-224: Hidden Symmetries in Simple Graphs]]>
http://www.mdpi.com/2073-8994/4/1/219
It is shown that each element s in the normalizer of the automorphism group Aut(G) of a simple graph G with labeled vertex set V is an Aut(G) invariant isomorphism between G and the graph obtained from G by the s permutation of V—i.e., s is a hidden permutation symmetry of G. A simple example illustrates the theory and the applied notion of system robustness for reconfiguration under symmetry constraint (RUSC) is introduced.Symmetry2012-03-0541Article10.3390/sym40102192192242073-89942012-03-05doi: 10.3390/sym4010219Allen D. Parks<![CDATA[Symmetry, Vol. 4, Pages 208-218: Self-Dual, Self-Petrie Covers of Regular Polyhedra]]>
http://www.mdpi.com/2073-8994/4/1/208
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.Symmetry2012-02-2741Article10.3390/sym40102082082182073-89942012-02-27doi: 10.3390/sym4010208Gabe Cunningham<![CDATA[Symmetry, Vol. 4, Pages 143-207: Intrinsic Symmetry Groups of Links with 8 and Fewer Crossings]]>
http://www.mdpi.com/2073-8994/4/1/143
We present an elementary derivation of the “intrinsic” symmetry groups for links of 8 or fewer crossings. We show that standard invariants are enough to rule out all potential symmetries outside the symmetry group of the group of the link for all but one of these links and present explicit isotopies generating the symmetry group for every link.Symmetry2012-02-2041Article10.3390/sym40101431432072073-89942012-02-20doi: 10.3390/sym4010143Michael BerglundJason CantarellaMeredith Perrie CaseyEleanor DannenbergWhitney GeorgeAja JohnsonAmelia KelleyAl LaPointeMatt MastinJason ParsleyJacob RooneyRachel Whitaker<![CDATA[Symmetry, Vol. 4, Pages 129-142: The 27 Possible Intrinsic Symmetry Groups of Two-Component Links]]>
http://www.mdpi.com/2073-8994/4/1/129
We consider the “intrinsic” symmetry group of a two-component link L, defined to be the image ∑(L) of the natural homomorphism from the standard symmetry group MCG(S3, L) to the product MCG(S3) × MCG(L). This group, first defined by Whitten in 1969, records directly whether L is isotopic to a link L′ obtained from L by permuting components or reversing orientations; it is a subgroup of Γ2, the group of all such operations. For two-component links, we catalog the 27 possible intrinsic symmetry groups, which represent the subgroups of Γ2 up to conjugacy. We are able to provide prime, nonsplit examples for 21 of these groups; some are classically known, some are new. We catalog the frequency at which each group appears among all 77,036 of the hyperbolic two-component links of 14 or fewer crossings in Thistlethwaite’s table. We also provide some new information about symmetry groups of the 293 non-hyperbolic two-component links of 14 or fewer crossings in the table.Symmetry2012-02-1741Article10.3390/sym40101291291422073-89942012-02-17doi: 10.3390/sym4010129Jason CantarellaJames CornishMatt MastinJason Parsley<![CDATA[Symmetry, Vol. 4, Pages 116-128: Defining the Symmetry of the Universal Semi-Regular Autonomous Asynchronous Systems]]>
http://www.mdpi.com/2073-8994/4/1/116
The regular autonomous asynchronous systems are the non-deterministic Boolean dynamical systems and universality means the greatest in the sense of the inclusion. The paper gives four definitions of symmetry of these systems in a slightly more general framework, called semi-regularity, and also many examples.Symmetry2012-02-1541Article10.3390/sym40101161161282073-89942012-02-15doi: 10.3390/sym4010116Serban E. Vlad<![CDATA[Symmetry, Vol. 4, Pages 39-115: Knots on a Torus: A Model of the Elementary Particles]]>
http://www.mdpi.com/2073-8994/4/1/39
Two knots; just two rudimentary knots, the unknot and the trefoil. That’s all we need to build a model of the elementary particles of physics, one with fermions and bosons, hadrons and leptons, interactions weak and strong and the attributes of spin, isospin, mass, charge, CPT invariance and more. There are no quarks to provide fractional charge, no gluons to sequester them within nucleons and no “colors” to avoid violating Pauli’s principle. Nor do we require the importation of an enigmatic Higgs boson to confer mass upon the particles of our world. All the requisite attributes emerge simply (and relativistically invariant) as a result of particle conformation and occupation in and of spacetime itself, a spacetime endowed with the imprimature of general relativity. Also emerging are some novel tools for systemizing the particle taxonomy as governed by the gauge group SU(2) and the details of particle degeneracy as well as connections to Hopf algebra, Dirac theory, string theory, topological quantum field theory and dark matter. One exception: it is found necessary to invoke the munificent geometry of the icosahedron in order to provide, as per the group “flavor” SU(3), a scaffold upon which to organize the well-known three generations—no more, no less—of the particle family tree.Symmetry2012-02-0941Article10.3390/sym4010039391152073-89942012-02-09doi: 10.3390/sym4010039Jack S. Avrin<![CDATA[Symmetry, Vol. 4, Pages 26-38: Symmetries of Spatial Graphs and Rational Twists along Spheres and Tori]]>
http://www.mdpi.com/2073-8994/4/1/26
A symmetry group of a spatial graph Γ in S3 is a finite group consisting of orientation-preserving self-diffeomorphisms of S3 which leave Γ setwise invariant. In this paper, we show that in many cases symmetry groups of Γ which agree on a regular neighborhood of Γ are equivalent up to conjugate by rational twists along incompressible spheres and tori in the exterior of Γ.Symmetry2012-01-2041Article10.3390/sym401002626382073-89942012-01-20doi: 10.3390/sym4010026Toru Ikeda<![CDATA[Symmetry, Vol. 4, Pages 15-25: Towards Symmetry-Based Explanation of (Approximate) Shapes of Alpha-Helices and Beta-Sheets (and Beta-Barrels) in Protein Structure]]>
http://www.mdpi.com/2073-8994/4/1/15
Protein structure is invariably connected to protein function. There are two important secondary structure elements: alpha-helices and beta-sheets (which sometimes come in a shape of beta-barrels). The actual shapes of these structures can be complicated, but in the first approximation, they are usually approximated by, correspondingly, cylindrical spirals and planes (and cylinders, for beta-barrels). In this paper, following the ideas pioneered by a renowned mathematician M. Gromov, we use natural symmetries to show that, under reasonable assumptions, these geometric shapes are indeed the best approximating families for secondary structures.Symmetry2012-01-1941Article10.3390/sym401001515252073-89942012-01-19doi: 10.3390/sym4010015Jaime NavaVladik Kreinovich<![CDATA[Symmetry, Vol. 4, Pages 1-14: Convex-Faced Combinatorially Regular Polyhedra of Small Genus]]>
http://www.mdpi.com/2073-8994/4/1/1
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.Symmetry2011-12-2841Article10.3390/sym40100011142073-89942011-12-28doi: 10.3390/sym4010001Egon SchulteJörg M. Wills<![CDATA[Symmetry, Vol. 3, Pages 828-851: Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings with Rotational Symmetry]]>
http://www.mdpi.com/2073-8994/3/4/828
We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have 3-, 4-, or 6-fold rotational symmetry. The symmetry groups of such tilings are of types p3, p31m, p4, p4g, and p6. There are no isohedral tilings with p3m1, p4m, or p6m symmetry groups that have polyominoes or polyiamonds as fundamental domains. We display the algorithms’ output and give enumeration tables for small values of n. This expands earlier works [1,2] and is a companion to [3].Symmetry2011-12-1234Article10.3390/sym30408288288512073-89942011-12-12doi: 10.3390/sym3040828Hiroshi FukudaChiaki KanomataNobuaki MutohGisaku NakamuraDoris Schattschneider<![CDATA[Symmetry, Vol. 3, Pages 780-827: An Application of the Extended Global SO(3) × SO(3) × U(1) Symmetry of the Hubbard Model on a Square Lattice: The Spinon, η-Spinon, and c Fermion Description]]>
http://www.mdpi.com/2073-8994/3/4/780
In this paper we review recent results on the preliminary applications of the new-found extended global SO(3) × SO(3) × U(1) symmetry of the Hubbard model on a bipartite lattice. Our results refer to the particular case of the bipartite square lattice. Specifically, we review a general description for such a model with nearest-neighbor transfer integral t and on-site repulsion U on a square lattice with N2a 1 sites consistent with its extended global symmetry. It refers to three types of elementary objects whose occupancy configurations generate the state representations of the model extended global symmetry. Such objects emerge from a suitable electron-rotated-electron unitary transformation. An application to the spin spectrum of the parent compound La2CuO4 is shortly reviewed.Symmetry2011-12-1234Article10.3390/sym30407807808272073-89942011-12-12doi: 10.3390/sym3040780Jose M. P. CarmeloMaria J. Sampaio<![CDATA[Symmetry, Vol. 3, Pages 767-779: Information Theory of Networks]]>
http://www.mdpi.com/2073-8994/3/4/767
The paper puts the emphasis on surveying information-theoretic network measures for analyzing the structure of networks. In order to apply the quantities interdisciplinarily, we also discuss some of their properties such as their structural interpretation and uniqueness.Symmetry2011-11-2934Article10.3390/sym30407677677792073-89942011-11-29doi: 10.3390/sym3040767Matthias Dehmer<![CDATA[Symmetry, Vol. 3, Pages 750-766: Symmetry in the Language of Gene Expression: A Survey of Gene Promoter Networks in Multiple Bacterial Species and Non-σ Regulons]]>
http://www.mdpi.com/2073-8994/3/4/750
The language of gene expression displays topological symmetry. An important step during gene expression is the binding of transcriptional proteins to DNA promoters adjacent to a gene. Some proteins bind to many promoters in a genome, defining a regulon of genes wherein each promoter might vary in DNA sequence relative to the average consensus. Here we examine the linguistic organization of gene promoter networks, wherein each node in the network represents a promoter and links between nodes represent the extent of base pair-sharing. Prior work revealed a fractal nucleus in several σ-factor regulons from Escherichia coli. We extend these findings to show fractal nuclei in gene promoter networks from three bacterial species, E. coli, Bacillus subtilis, and Pseudomonas aeruginosa. We surveyed several non-σ transcription factors from these species and found that many contain a nucleus that is both visually and numerically fractal. Promoter footprint size scaled as a negative power-law with both information entropy and fractal dimension, while the latter two parameters scaled positively and linearly. The fractal dimension of the diffuse networks (dB = ~1.7) was close to that expected of a diffusion limited aggregation process, confirming prior predictions as to a possible mechanism for development of this structure.Symmetry2011-11-1834Article10.3390/sym30407507507662073-89942011-11-18doi: 10.3390/sym3040750Preston R. AldrichRobert K. HorsleyStefan M. Turcic<![CDATA[Symmetry, Vol. 3, Pages 699-749: d-Wave Superconductivity and s-Wave Charge Density Waves: Coexistence between Order Parameters of Different Origin and Symmetry]]>
http://www.mdpi.com/2073-8994/3/4/699
A review of the theory describing the coexistence between d-wave superconductivity and s-wave charge-density-waves (CDWs) is presented. The CDW gapping is identified with pseudogapping observed in high-Tc oxides. According to the cuprate specificity, the analysis is carried out for the two-dimensional geometry of the Fermi surface (FS). Phase diagrams on the σ0 − α plane—here, σ0 is the ratio between the energy gaps in the parent pure CDW and superconducting states, and the quantity 2α is connected with the degree of dielectric (CDW) FS gapping—were obtained for various possible configurations of the order parameters in the momentum space. Relevant tunnel and photoemission experimental data for high-Tc oxides are compared with theoretical predictions. A brief review of the results obtained earlier for the coexistence between s-wave superconductivity and CDWs is also given.Symmetry2011-10-2034Article10.3390/sym30406996997492073-89942011-10-20doi: 10.3390/sym3040699Toshikazu EkinoAlexander M. GabovichMai Suan LiMarek PękałaHenryk SzymczakAlexander I. Voitenko<![CDATA[Symmetry, Vol. 3, Pages 680-698: Symmetry and Evidential Support]]>
http://www.mdpi.com/2073-8994/3/3/680
This article proves that formal theories of evidential favoring must fail because they are inevitably language dependent. I begin by describing Carnap’s early confirmation theories to show how language dependence problems (like Goodman’s grue problem) arise. I then generalize to showthat any formal favoring theory satisfying minimal plausible conditions will yield different judgments about the same evidence and hypothesis when they are expressed in alternate languages. This does not just indict formal theories of favoring; it also shows that something beyond our evidence must be invoked to substantively favor one hypothesis over another.Symmetry2011-09-1633Article10.3390/sym30306806806982073-89942011-09-16doi: 10.3390/sym3030680Michael G. Titelbaum<![CDATA[Symmetry, Vol. 3, Pages 653-679: Lattices of Graphical Gaussian Models with Symmetries]]>
http://www.mdpi.com/2073-8994/3/3/653
In order to make graphical Gaussian models a viable modelling tool when the number of variables outgrows the number of observations, [1] introduced model classes which place equality restrictions on concentrations or partial correlations. The models can be represented by vertex and edge coloured graphs. The need for model selection methods makes it imperative to understand the structure of model classes. We identify four model classes that form complete lattices of models with respect to model inclusion, which qualifies them for an Edwards–Havránek model selection procedure [2]. Two classes turn out most suitable for a corresponding model search. We obtain an explicit search algorithm for one of them and provide a model search example for the other.Symmetry2011-09-0733Article10.3390/sym30306536536792073-89942011-09-07doi: 10.3390/sym3030653Helene Gehrmann<![CDATA[Symmetry, Vol. 3, Pages 636-652: Symmetry and the Brown-Freiling Refutation of the Continuum Hypothesis]]>
http://www.mdpi.com/2073-8994/3/3/636
Freiling [1] and Brown [2] have put forward a probabilistic reductio argument intended to refute the Continuum Hypothesis. The argument relies heavily upon intuitions about symmetry in a particular scenario. This paper argues that the argument fails, but is still of interest for two reasons. First, the failure is unusual in that the symmetry intuitions are demonstrably coherent, even though other constraints make it impossible to find a probability model for the scenario. Second, the best probability models have properties analogous to non-conglomerability, motivating a proposed extension of that concept (and corresponding limits on Bayesian conditionalization).Symmetry2011-09-0633Article10.3390/sym30306366366522073-89942011-09-06doi: 10.3390/sym3030636Paul Bartha<![CDATA[Symmetry, Vol. 3, Pages 611-635: Symmetry, Invariance and Ontology in Physics and Statistics]]>
http://www.mdpi.com/2073-8994/3/3/611
This paper has three main objectives: (a) Discuss the formal analogy between some important symmetry-invariance arguments used in physics, probability and statistics. Specifically, we will focus on Noether’s theorem in physics, the maximum entropy principle in probability theory, and de Finetti-type theorems in Bayesian statistics; (b) Discuss the epistemological and ontological implications of these theorems, as they are interpreted in physics and statistics. Specifically, we will focus on the positivist (in physics) or subjective (in statistics) interpretations vs. objective interpretations that are suggested by symmetry and invariance arguments; (c) Introduce the cognitive constructivism epistemological framework as a solution that overcomes the realism-subjectivism dilemma and its pitfalls. The work of the physicist and philosopher Max Born will be particularly important in our discussion.Symmetry2011-09-0133Article10.3390/sym30306116116352073-89942011-09-01doi: 10.3390/sym3030611Julio Michael Stern<![CDATA[Symmetry, Vol. 3, Pages 600-610: High-Dimensional Random Matrices from the Classical Matrix Groups, and Generalized Hypergeometric Functions of Matrix Argument]]>
http://www.mdpi.com/2073-8994/3/3/600
Results from the theory of the generalized hypergeometric functions of matrix argument, and the related zonal polynomials, are used to develop a new approach to study the asymptotic distributions of linear functions of uniformly distributed random matrices from the classical compact matrix groups. In particular, we provide a new approach for proving the following result of D’Aristotile, Diaconis, and Newman: Let the random matrix Hn be uniformly distributed according to Haar measure on the group of n × n orthogonal matrices, and let An be a non-random n × n real matrix such that tr (A'nAn) = 1. Then, as n→∞, √n tr AnHn converges in distribution to the standard normal distribution.Symmetry2011-08-2633Article10.3390/sym30306006006102073-89942011-08-26doi: 10.3390/sym3030600Donald St. P. Richards<![CDATA[Symmetry, Vol. 3, Pages 574-599: Squaring the Circle and Cubing the Sphere: Circular and Spherical Copulas]]>
http://www.mdpi.com/2073-8994/3/3/574
Do there exist circular and spherical copulas in ℝd? That is, do there exist circularly symmetric distributions on the unit disk in ℝ2 and spherically symmetric distributions on the unit ball in ℝd, d ≥ 3, whose one-dimensional marginal distributions are uniform? The answer is yes for d = 2 and 3, where the circular and spherical copulas are unique and can be determined explicitly, but no for d ≥ 4. A one-parameter family of elliptical bivariate copulas is obtained from the unique circular copula in ℝ2 by oblique coordinate transformations. Copulas obtained by a non-linear transformation of a uniform distribution on the unit ball in ℝd are also described, and determined explicitly for d = 2.Symmetry2011-08-2333Article10.3390/sym30305745745992073-89942011-08-23doi: 10.3390/sym3030574Michael D. PerlmanJon A. Wellner<![CDATA[Symmetry, Vol. 3, Pages 564-573: Green’s Symmetries in Finite Digraphs]]>
http://www.mdpi.com/2073-8994/3/3/564
The semigroup DV of digraphs on a set V of n labeled vertices is defined. It is shown that DV is faithfully represented by the semigroup Bn of n ´ n Boolean matrices and that the Green’s L, R, H, and D equivalence classifications of digraphs in DV follow directly from the Green’s classifications already established for Bn. The new results found from this are: (i) L, R, and H equivalent digraphs contain sets of vertices with identical neighborhoods which remain invariant under certain one-sided semigroup multiplications that transform one digraph into another within the same equivalence class, i.e., these digraphs exhibit Green’s isoneighborhood symmetries; and (ii) D equivalent digraphs are characterized by isomorphic inclusion lattices that are generated by their out-neighborhoods and which are preserved under certain two-sided semigroup multiplications that transform digraphs within the same D equivalence class, i.e., these digraphs are characterized by Green’s isolattice symmetries. As a simple illustrative example, the Green’s classification of all digraphs on two vertices is presented and the associated Green’s symmetries are identified.Symmetry2011-08-1533Article10.3390/sym30305645645732073-89942011-08-15doi: 10.3390/sym3030564Allen D. Parks<![CDATA[Symmetry, Vol. 3, Pages 541-563: Symmetry Aspects of the Band Structure and Motion Equations Applied in Calculating the Cyclotron Frequency of Electrons in Metals]]>
http://www.mdpi.com/2073-8994/3/3/541
Cyclotron frequency of a crystal electron is, in general, not an easily accessible parameter. Nevertheless, its calculation can be simplified when the symmetry properties of the band structure and those of the motion equations in the magnetic field are simultaneously taken into account. In effect, a combined symmetry of the electron Hamiltonian and that of the Lorentz equation provide us with a non-linear oscillator problem of high symmetry. In the next step, the kinetic energy of the oscillator can be separated from the whole of electron energy and applied in a new kind of calculation of the cyclotron frequency which is much more simple than before. In consequence, a detailed approach to the electron circulation, also in more complex band structures, becomes a relatively easy task. For different crystal lattices of cubic symmetry taken as examples the cyclotron frequency of the present and a former method are compared numerically giving the same results.Symmetry2011-08-1033Article10.3390/sym30305415415632073-89942011-08-10doi: 10.3390/sym3030541Stanislaw OlszewskiTomasz Roliński<![CDATA[Symmetry, Vol. 3, Pages 524-540: Action Duality: A Constructive Principle for Quantum Foundations]]>
http://www.mdpi.com/2073-8994/3/3/524
An analysis of the path integral approach to quantum theory motivates the hypothesis that two experiments with the same classical action should have dual ontological descriptions. If correct, this hypothesis would not only constrain realistic interpretations of quantum theory, but would also act as a constructive principle, allowing any realistic model of one experiment to generate a corresponding model for its action-dual. Two pairs of action-dual experiments are presented, including one experiment that violates the Bell inequality and yet is action-dual to a single particle. The implications generally support retrodictive and retrocausal interpretations.Symmetry2011-07-2733Article10.3390/sym30305245245402073-89942011-07-27doi: 10.3390/sym3030524Ken B. WhartonDavid J. MillerHuw Price<![CDATA[Symmetry, Vol. 3, Pages 503-523: Folded Sheet Versus Transparent Sheet Models for Human Symmetry Judgments]]>
http://www.mdpi.com/2073-8994/3/3/503
As a contribution to the mysteries of human symmetry perception, reaction time data were collected on the detection of symmetry or repetition violations, in the context of short term visual memory studies. The histograms for reaction time distributions are rather narrow in the case of symmetry judgments. Their analysis was performed in terms of a simple kinetic model of a mental process in two steps, a slow one for the construction of the representation of the images to be compared, and a fast one, in the 50 ms range, for the decision. There was no need for an additional ‘mental rotation’ step. Symmetry seems to facilitate the construction step. I also present here original stimuli showing a color equalization effect across a symmetry axis, and its counterpart in periodic patterns. According to a “folded sheet model”, when a shape is perceived, the brain automatically constructs a mirror-image representation of the shape. Based in part on the reaction time analysis, I present here an alternative “transparent sheet” model in which the brain constructs a single representation, which can be accessed from two sides, thus generating simultaneously a pattern and its mirror-symmetric partner. Filtering processes, implied by current models of symmetry perception could intervene at an early stage, by nucleating the propagation of similar perceptual groupings in the two symmetric images.Symmetry2011-07-2233Article10.3390/sym30305035035232073-89942011-07-22doi: 10.3390/sym3030503Jacques Ninio<![CDATA[Symmetry, Vol. 3, Pages 487-502: Classifying Entropy Measures]]>
http://www.mdpi.com/2073-8994/3/3/487
Our paper analyzes some aspects of Uncertainty Measures. We need to obtain new ways to model adequate conditions or restrictions, constructed from vague pieces of information. The classical entropy measure originates from scientific fields; more specifically, from Statistical Physics and Thermodynamics. With time it was adapted by Claude Shannon, creating the current expanding Information Theory. However, the Hungarian mathematician, Alfred Rényi, proves that different and valid entropy measures exist in accordance with the purpose and/or need of application. Accordingly, it is essential to clarify the different types of measures and their mutual relationships. For these reasons, we attempt here to obtain an adequate revision of such fuzzy entropy measures from a mathematical point of view.Symmetry2011-07-2033Article10.3390/sym30304874875022073-89942011-07-20doi: 10.3390/sym3030487Angel Garrido<![CDATA[Symmetry, Vol. 3, Pages 472-486: On Symmetry of Independence Polynomials]]>
http://www.mdpi.com/2073-8994/3/3/472
An independent set in a graph is a set of pairwise non-adjacent vertices, and α(G) is the size of a maximum independent set in the graph G. A matching is a set of non-incident edges, while μ(G) is the cardinality of a maximum matching. If sk is the number of independent sets of size k in G, then I(G; x) = s0 + s1x + s2x2 + ... + sαxα, α = α (G), is called the independence polynomial of G (Gutman and Harary, 1986). If sj = sαj for all 0 ≤ j ≤ [α/2], then I(G; x) is called symmetric (or palindromic). It is known that the graph G ° 2K1, obtained by joining each vertex of G to two new vertices, has a symmetric independence polynomial (Stevanović, 1998). In this paper we develop a new algebraic technique in order to take care of symmetric independence polynomials. On the one hand, it provides us with alternative proofs for some previously known results. On the other hand, this technique allows to show that for every graph G and for each non-negative integer k ≤ μ (G), one can build a graph H, such that: G is a subgraph of H, I (H; x) is symmetric, and I (G ° 2K1; x) = (1 + x)k · I (H; x).Symmetry2011-07-1533Article10.3390/sym30304724724862073-89942011-07-15doi: 10.3390/sym3030472Vadim E. LevitEugen Mandrescu<![CDATA[Symmetry, Vol. 3, Pages 457-471: Mirror Symmetry Is Subject to Crowding]]>
http://www.mdpi.com/2073-8994/3/3/457
Mirror symmetry is often thought to be particularly salient to human observers because it engages specialized mechanisms that evolved to sense symmetrical objects in nature. Although symmetry is indeed present in many of our artifacts and markings on wildlife, studies have shown that sensitivity to mirror symmetry does not serve an alerting function and sensitivity to symmetry decreases in a rather unremarkable way when it is presented away from the centre of the visual field. Here we show that symmetrical targets are vulnerable to the same interference as other stimuli when surrounded by non-target elements. These results provide further evidence that symmetry is not special to the early visual system, and reinforce the notion that our fascination with symmetry is more likely attributable to cognitive and aesthetic factors than to specialized, low level mechanisms in the visual system.Symmetry2011-07-1333Article10.3390/sym30304574574712073-89942011-07-13doi: 10.3390/sym3030457Gabrielle RoddyRick Gurnsey<![CDATA[Symmetry, Vol. 3, Pages 443-456: Reduction of Image Complexity Explains Aesthetic Preference for Symmetry]]>
http://www.mdpi.com/2073-8994/3/3/443
Symmetric patterns are more appealing to human observers than asymmetric ones. Here, we investigate the visual information processing mechanisms underlying this aesthetic preference. All stimuli were derived from phase scrambled versions of forty face or nature images. In addition to the scrambled images, there were four other types of test image: symmetric, in which one part of the image was a reflection of another around an axis; repetitive, in which one part of the image was a copy of the other; anti-symmetric, similar to symmetric but with the contrast of one side reversed; and interleaved patterns, in which half of the symmetric pattern was replaced by a scrambled image. The number of axes ranged from 1 to 16 for all image types. The task of our 20 observers was to give a preference rating to each image on a 6-point Lickert scale. The preference rating increased with the number of axes for all stimulus types. The observers showed a similar preference for symmetric and repetitive patterns and slightly less preference for anti-symmetric patterns. The preference for interleaved patterns was much less than for other types of stimuli. Preference for an image cannot be explained by either the ecological significance of its content or the slope of its amplitude spectrum. Instead, preference can be accounted for by the complexity of the image.Symmetry2011-07-1133Article10.3390/sym30304434434562073-89942011-07-11doi: 10.3390/sym3030443Chien-Chung ChenJo-Hsuan WuChia-Ching Wu<![CDATA[Symmetry, Vol. 3, Pages 402-442: Linear Recurrent Double Sequences with Constant Border in M2(F2) are Classified According to Their Geometric Content]]>
http://www.mdpi.com/2073-8994/3/3/402
The author used the automatic proof procedure introduced in [1] and verified that the 4096 homomorphic recurrent double sequences with constant borders defined over Klein’s Vierergruppe K and the 4096 linear recurrent double sequences with constant border defined over the matrix ring M2(F2) can be also produced by systems of substitutions with finitely many rules. This permits the definition of a sound notion of geometric content for most of these sequences, more exactly for those which are not primitive. We group the 4096 many linear recurrent double sequences with constant border I over the ring M2(F2) in 90 geometric types. The classification over Klein’s Vierergruppe Kis not explicitly displayed and consists of the same geometric types like for M2(F2), but contains more exceptions. There are a lot of cases of unsymmetric double sequences converging to symmetric geometric contents. We display also geometric types occurring both in a monochromatic and in a dichromatic version.Symmetry2011-07-0733Article10.3390/sym30304024024422073-89942011-07-07doi: 10.3390/sym3030402Mihai Prunescu<![CDATA[Symmetry, Vol. 3, Pages 389-401: Is the Notion of Time Really Fundamental?]]>
http://www.mdpi.com/2073-8994/3/3/389
From the physics point of view, time is now best described through General Relativity as part of space-time, which is a dynamical object encoding gravity. Time possesses also some intrinsic irreversibility due to thermodynamics and quantum mechanical effects. This irreversibility can look puzzling since time-like loops (and hence time machines) can appear in General Relativity (for example in the Gödel universe, a solution of Einstein’s equations). We take this apparent discrepancy as a warning bell, pointing out that time as we understand it might not be fundamental and that whatever theory lying beyond General Relativity may not include time as we know it as a fundamental structure. We propose therefore, following the philosophy of analog models of gravity, that time and gravity might not be fundamental per se, but only emergent features. We illustrate our proposal using a toy-model where we show how the Lorentzian signature and Nordström gravity (a diffeomorphisms invariant scalar gravity theory) can emerge from a timeless non-dynamical space. This article received the fourth prize at the essay competition of the Foundational Questions Institute on the nature of time.Symmetry2011-06-2933Article10.3390/sym30303893894012073-89942011-06-29doi: 10.3390/sym3030389Florian GirelliStefano LiberatiLorenzo Sindoni<![CDATA[Symmetry, Vol. 3, Pages 365-388: Any Pair of 2D Curves Is Consistent with a 3D Symmetric Interpretation]]>
http://www.mdpi.com/2073-8994/3/2/365
Symmetry has been shown to be a very effective a priori constraint in solving a 3D shape recovery problem. Symmetry is useful in 3D recovery because it is a form of redundancy. There are, however, some fundamental limits to the effectiveness of symmetry. Specifically, given two arbitrary curves in a single 2D image, one can always find a 3D mirror-symmetric interpretation of these curves under quite general assumptions. The symmetric interpretation is unique under a perspective projection and there is a one parameter family of symmetric interpretations under an orthographic projection. We formally state and prove this observation for the case of one-to-one and many-to-many point correspondences. We conclude by discussing the role of degenerate views, higher-order features in determining the point correspondences, as well as the role of the planarity constraint. When the correspondence of features is known and/or curves can be assumed to be planar, 3D symmetry becomes non-accidental in the sense that a 2D image of a 3D asymmetric shape obtained from a random viewing direction will not allow for 3D symmetric interpretations.Symmetry2011-06-1032Article10.3390/sym30203653653882073-89942011-06-10doi: 10.3390/sym3020365Tadamasa SawadaYunfeng LiZygmunt Pizlo<![CDATA[Symmetry, Vol. 3, Pages 325-364: Polyominoes and Polyiamonds as Fundamental Domains for Isohedral Tilings of Crystal Class D2]]>
http://www.mdpi.com/2073-8994/3/2/325
We describe computer algorithms that produce the complete set of isohedral tilings by n-omino or n-iamond tiles in which the tiles are fundamental domains and the tilings have pmm, pmg, pgg or cmm symmetry [1]. These symmetry groups are members of the crystal class D2 among the 17 two-dimensional symmetry groups [2]. We display the algorithms’ output and give enumeration tables for small values of n. This work is a continuation of our earlier works for the symmetry groups p3, p31m, p3m1, p4, p4g, p4m, p6, and p6m [3–5].Symmetry2011-06-0932Article10.3390/sym30203253253642073-89942011-06-09doi: 10.3390/sym3020325Hiroshi FukudaChiaki KanomataNobuaki MutohGisaku NakamuraDoris Schattschneider<![CDATA[Symmetry, Vol. 3, Pages 305-324: Symmetry Groups for the Decomposition of Reversible Computers, Quantum Computers, and Computers in between]]>
http://www.mdpi.com/2073-8994/3/2/305
Whereas quantum computing circuits follow the symmetries of the unitary Lie group, classical reversible computation circuits follow the symmetries of a finite group, i.e., the symmetric group. We confront the decomposition of an arbitrary classical reversible circuit with w bits and the decomposition of an arbitrary quantum circuit with w qubits. Both decompositions use the control gate as building block, i.e., a circuit transforming only one (qu)bit, the transformation being controlled by the other w−1 (qu)bits. We explain why the former circuit can be decomposed into 2w − 1 control gates, whereas the latter circuit needs 2w − 1 control gates. We investigate whether computer circuits, not based on the full unitary group but instead on a subgroup of the unitary group, may be decomposable either into 2w − 1 or into 2w − 1 control gates.Symmetry2011-06-0732Article10.3390/sym30203053053242073-89942011-06-07doi: 10.3390/sym3020305Alexis De VosStijn De Baerdemacker<![CDATA[Symmetry, Vol. 3, Pages 283-304: Enriching the Symmetry of Maxwell Equations through Unprecedented Magnetic Responses of Artificial Metamaterials and Their Revolutionary Applications]]>
http://www.mdpi.com/2073-8994/3/2/283
The major issue regarding magnetic response in nature—“negative values for the permeability μ of material parameters, especially in terahertz or optical region” makes the electromagnetic properties of natural materials asymmetric. Recently, research in metamaterials has grown in significance because these artificial materials can demonstrate special and, indeed, extraordinary electromagnetic phenomena such as the inverse of Snell’s law and novel applications. A critical topic in metamaterials is the artificial negative magnetic response, which can be designed in the higher frequency regime (from microwave to optical range). Artificial magnetism illustrates new physics and new applications, which have been demonstrated over the past few years. In this review, we present recent developments in research on artificial magnetic metamaterials including split-ring resonator structures, sandwich structures, and high permittivity-based dielectric composites. Engineering applications such as invisibility cloaking, negative refractive index medium, and slowing light fall into this category. We also discuss the possibility that metamaterials can be suitable for realizing new and exotic electromagnetic properties.Symmetry2011-06-0332Review10.3390/sym30202832833042073-89942011-06-03doi: 10.3390/sym3020283Yueh-Chun LaiCheng-Kuang ChenTsung-Yu HuangIeng-Wai UnYu-Hang YangTa-Jen Yen<![CDATA[Symmetry, Vol. 3, Pages 265-282: Prolinethioamides versus Prolinamides in Organocatalyzed Aldol Reactions—A Comparative Study]]>
http://www.mdpi.com/2073-8994/3/2/265
Various organocatalysts have been developed for the aldol reaction but particular attention has been paid to prolinamide derivatives. They are easy to prepare and their catalytic activity can be readily tuned through structural modification. In this review, the comparison of catalytic activities between prolinethioamides and their respective amides in direct asymmetric aldol reactions is presented.Symmetry2011-06-0132Review10.3390/sym30202652652822073-89942011-06-01doi: 10.3390/sym3020265Dorota GrykoMikołaj ChromińskiDominika J. Pielacińska<![CDATA[Symmetry, Vol. 3, Pages 246-264: Similar Symmetries: The Role of Wallpaper Groups in Perceptual Texture Similarity]]>
http://www.mdpi.com/2073-8994/3/2/246
Periodic patterns and symmetries are striking visual properties that have been used decoratively around the world throughout human history. Periodic patterns can be mathematically classified into one of 17 different Wallpaper groups, and while computational models have been developed which can extract an image's symmetry group, very little work has been done on how humans perceive these patterns. This study presents the results from a grouping experiment using stimuli from the different wallpaper groups. We find that while different images from the same wallpaper group are perceived as similar to one another, not all groups have the same degree of self-similarity. The similarity relationships between wallpaper groups appear to be dominated by rotations.Symmetry2011-05-2532Article10.3390/sym30202462462642073-89942011-05-25doi: 10.3390/sym3020246Alasdair D. F. ClarkePatrick R. GreenFraser HalleyMike J. Chantler<![CDATA[Symmetry, Vol. 3, Pages 220-245: Organocatalytic Enantioselective Henry Reactions]]>
http://www.mdpi.com/2073-8994/3/2/220
A large number of interesting organocatalytic enantioselective protocols have been explored and successfully applied in the last decade. Among them, the Henry (nitroaldol) reaction represents a powerful carbon-carbon bond-forming procedure for the preparation of valuable synthetic intermediates, such as enantioenriched nitro alcohols, which can be further transformed in a number of important nitrogen and oxygen-containing compounds. This area of research is still in expansion and a more complex version of this useful process has recently emerged, the domino Michael/Henry protocol, affording highly functionalized cycles with multiple stereogenic centers.Symmetry2011-05-2332Review10.3390/sym30202202202452073-89942011-05-23doi: 10.3390/sym3020220Yolanda Alvarez-CasaoEugenia Marques-LopezRaquel P. Herrera<![CDATA[Symmetry, Vol. 3, Pages 207-219: Visual Discrimination of the 17 Plane Symmetry Groups]]>
http://www.mdpi.com/2073-8994/3/2/207
Within most of the 17 plane symmetry groups, individual symmetry operations act in multiple, nonequivalent ways. This, and the fact that many groups can be realized on the basis of different unit cells and generating regions, poses difficulties for visual discrimination and identification. Because of inherent confounds, only few of the groups can be studied by traditional experimental methodology. The use of an oddity paradigm and specific tiling patterns that camouflage groups in complex textures are recommended as partial remedy to this impasse. In order to prepare readers for an appreciation of the aforementioned issues and to provide a rationale for their investigation, the reporting of experiments and the discussion of methodological problems is preceded by a brief overview of the role which symmetry has played in the visual arts.Symmetry2011-05-1132Article10.3390/sym30202072072192073-89942011-05-11doi: 10.3390/sym3020207Klaus Landwehr<![CDATA[Symmetry, Vol. 3, Pages 171-206: Quantum Theory and Probability Theory: Their Relationship and Origin in Symmetry]]>
http://www.mdpi.com/2073-8994/3/2/171
Quantum theory is a probabilistic calculus that enables the calculation of the probabilities of the possible outcomes of a measurement performed on a physical system. But what is the relationship between this probabilistic calculus and probability theory itself? Is quantum theory compatible with probability theory? If so, does it extend or generalize probability theory? In this paper, we answer these questions, and precisely determine the relationship between quantum theory and probability theory, by explicitly deriving both theories from first principles. In both cases, the derivation depends upon identifying and harnessing the appropriate symmetries that are operative in each domain. We prove, for example, that quantum theory is compatible with probability theory by explicitly deriving quantum theory on the assumption that probability theory is generally valid.Symmetry2011-04-2732Article10.3390/sym30201711712062073-89942011-04-27doi: 10.3390/sym3020171Philip GoyalKevin H. Knuth<![CDATA[Symmetry, Vol. 3, Pages 165-170: Facile and Convenient One-Pot Process for the Synthesis of Spirooxindole Derivatives in High Optical Purity Using (−)-(S)-Brevicolline as an Organocatalyst]]>
http://www.mdpi.com/2073-8994/3/2/165
The paper presents an application of the asymmetry approach to spirooxindoles via Brevicolline, Cinchonidine or Cinchonine catalyzed one-pot multicomponent synthesis. Brevicolline, in comparison with Cinchonidine or Cinchonine, catalyzes the reaction of isatins, acetylacetone/ethyl 3-oxobutanoate and malononitrile, with the formation of spiro[oxindole-3,4'-4'H-pirane] derivatives in an optically active form in very good to excellent yields.Symmetry2011-04-2032Article10.3390/sym30201651651702073-89942011-04-20doi: 10.3390/sym3020165Fliur MacaevNatalia SucmanFelix ShepeliMarina ZveaghintsevaVsevolod Pogrebnoi<![CDATA[Symmetry, Vol. 3, Pages 155-164: Primary Amino Acid Lithium Salt-Catalyzed Asymmetric Michael Addition of Carbon Nucleophiles to Enones]]>
http://www.mdpi.com/2073-8994/3/2/155
Asymmetric Michael addition of carbon nucleophiles, nitroalkanes and a β-ketoester, to enones was investigated by using a primary amino acid lithium salt as a catalyst.Symmetry2011-04-0832Article10.3390/sym30201551551642073-89942011-04-08doi: 10.3390/sym3020155Masanori YoshidaKeisuke HiramaMao NaritaShoji Hara<![CDATA[Symmetry, Vol. 3, Pages 134-154: Quantisation, Representation and Reduction; How Should We Interpret the Quantum Hamiltonian Constraints of Canonical Gravity?]]>
http://www.mdpi.com/2073-8994/3/2/134
Hamiltonian constraints feature in the canonical formulation of general relativity. Unlike typical constraints they cannot be associated with a reduction procedure leading to a non-trivial reduced phase space and this means the physical interpretation of their quantum analogues is ambiguous. In particular, can we assume that “quantisation commutes with reduction” and treat the promotion of these constraints to operators annihilating the wave function, according to a Dirac type procedure, as leading to a Hilbert space equivalent to that reached by quantisation of the problematic reduced space? If not, how should we interpret Hamiltonian constraints quantum mechanically? And on what basis do we assert that quantisation and reduction commute anyway? These questions will be refined and explored in the context of modern approaches to the quantisation of canonical general relativity.Symmetry2011-03-3132Article10.3390/sym30201341341542073-89942011-03-31doi: 10.3390/sym3020134Karim P. Y. Thébault<![CDATA[Symmetry, Vol. 3, Pages 126-133: Monochrome Symmetric Subsets in Colorings of Finite Abelian Groups]]>
http://www.mdpi.com/2073-8994/3/2/126
A subset S of a group G is symmetric if there is an element g є G such that gS-1g = S. We study some Ramsey type functions for symmetric subsets in finite Abelian groups.Symmetry2011-03-2432Article10.3390/sym30201261261332073-89942011-03-24doi: 10.3390/sym3020126Yuliya Zelenyuk<![CDATA[Symmetry, Vol. 3, Pages 84-125: Asymmetric Organocatalytic Reactions of α,β-Unsaturated Cyclic Ketones]]>
http://www.mdpi.com/2073-8994/3/1/84
The 1,4-conjugate addition of nucleophiles to α,β-unsaturated carbonyl compounds represents one fundamental bond-forming reaction in organic synthesis. The development of effective organocatalysts for the enantioselective conjugate addition of malonate, nitroalkane and other carbon and heteroatom nucleophiles to cycloenones constitutes an important research field and has been explored in recent years. At the same time, asymmetric Diels-Alder reactions have been developed and often a mechanism has been demonstrated to be a double addition rather than synchronous. This review aims to cover literature up to the end of 2010, describing all the different organocatalytic asymmetric 1,4-conjugate additions even if they are listed as transfer hydrogenation, cycloadditions or desymmetrization of aromatic compounds.Symmetry2011-03-2231Review10.3390/sym3010084841252073-89942011-03-22doi: 10.3390/sym3010084Renato DalpozzoGiuseppe BartoliGiorgio Bencivenni<![CDATA[Symmetry, Vol. 3, Pages 72-83: Long Time Behaviour on a Path Group of the Heat Semi-group Associated to a Bilaplacian]]>
http://www.mdpi.com/2073-8994/3/1/72
We show that in long-time the heat semi-group on a path group associated to a Bilaplacian on the group tends to the Haar distribution on a path group.Symmetry2011-03-2131Article10.3390/sym301007272832073-89942011-03-21doi: 10.3390/sym3010072Remi Leandre<![CDATA[Symmetry, Vol. 3, Pages 54-71: The Influence of Perception on the Distribution of Multiple Symmetries in Nature and Art]]>
http://www.mdpi.com/2073-8994/3/1/54
Much is already known about single mirror symmetry, but multiple mirror symmetry is still understood poorly. In particular, perceptually, multiple symmetry does not seem to behave as suggested by the number of symmetry axes alone. Here, theoretical ideas on single symmetry perception and their extensions to multiple symmetry are discussed alongside empirical findings on multiple symmetry perception. The evidence suggests that, apart from the number of axes, also their relative orientation is perceptually relevant. This, in turn, suggests that perception is responsible for the preponderance of 3-fold and 5-fold symmetries in flowers as well as for their absence in decorative art.Symmetry2011-03-2131Article10.3390/sym301005454712073-89942011-03-21doi: 10.3390/sym3010054Peter A. Van der Helm<![CDATA[Symmetry, Vol. 3, Pages 37-53: The First Appearance of Symmetry in the Human Lineage: Where Perception Meets Art]]>
http://www.mdpi.com/2073-8994/3/1/37
Although symmetry may be important for understanding the selection of form in art over the historical period, this preference may have originally stemmed from certain basic perceptual mechanism that initially arose during prehistory. The first signs of an awareness to symmetry can be found in the archaeological record with the arrival of Acheulean handaxes, especially those dating from 500,000 years ago onwards, which are typified by a prodigious bilateral symmetry. As handaxes represent the earliest material record of an interest in symmetry by the human lineage, they provide a privileged means of understanding why this kind of form came to be valued by later human groups, particularly in relation to “art”. Although still controversial, the preference for symmetry at such an early date has been linked to various aspects of perception relating to enduring evolutionary factors. In this regard, it will be demonstrated how the preference for symmetrical Acheulean tools arose out of long standing perceptual correlates relating to ecological factors that predated the arrival of hominins.Symmetry2011-03-0131Review10.3390/sym301003737532073-89942011-03-01doi: 10.3390/sym3010037Derek Hodgson<![CDATA[Symmetry, Vol. 3, Pages 16-36: Lorentz Harmonics, Squeeze Harmonics and Their Physical Applications]]>
http://www.mdpi.com/2073-8994/3/1/16
Among the symmetries in physics, the rotation symmetry is most familiar to us. It is known that the spherical harmonics serve useful purposes when the world is rotated. Squeeze transformations are also becoming more prominent in physics, particularly in optical sciences and in high-energy physics. As can be seen from Dirac’s light-cone coordinate system, Lorentz boosts are squeeze transformations. Thus the squeeze transformation is one of the fundamental transformations in Einstein’s Lorentz-covariant world. It is possible to define a complete set of orthonormal functions defined for one Lorentz frame. It is shown that the same set can be used for other Lorentz frames. Transformation properties are discussed. Physical applications are discussed in both optics and high-energy physics. It is shown that the Lorentz harmonics provide the mathematical basis for squeezed states of light. It is shown also that the same set of harmonics can be used for understanding Lorentz-boosted hadrons in high-energy physics. It is thus possible to transmit physics from one branch of physics to the other branch using the mathematical basis common to them.Symmetry2011-02-1431Review10.3390/sym301001616362073-89942011-02-14doi: 10.3390/sym3010016Young S. KimMarilyn E. Noz<![CDATA[Symmetry, Vol. 3, Pages 1-15: Symmetry in Complex Networks]]>
http://www.mdpi.com/2073-8994/3/1/1
In this paper, we analyze a few interrelated concepts about graphs, such as their degree, entropy, or their symmetry/asymmetry levels. These concepts prove useful in the study of different types of Systems, and particularly, in the analysis of Complex Networks. A System can be defined as any set of components functioning together as a whole. A systemic point of view allows us to isolate a part of the world, and so, we can focus on those aspects that interact more closely than others. Network Science analyzes the interconnections among diverse networks from different domains: physics, engineering, biology, semantics, and so on. Current developments in the quantitative analysis of Complex Networks, based on graph theory, have been rapidly translated to studies of brain network organization. The brain's systems have complex network features—such as the small-world topology, highly connected hubs and modularity. These networks are not random. The topology of many different networks shows striking similarities, such as the scale-free structure, with the degree distribution following a Power Law. How can very different systems have the same underlying topological features? Modeling and characterizing these networks, looking for their governing laws, are the current lines of research. So, we will dedicate this Special Issue paper to show measures of symmetry in Complex Networks, and highlight their close relation with measures of information and entropy.Symmetry2011-01-1031Article10.3390/sym30100011152073-89942011-01-10doi: 10.3390/sym3010001Angel Garrido<![CDATA[Symmetry, Vol. 2, Pages 1945-1980: Positive Cosmological Constant and Quantum Theory]]>
http://www.mdpi.com/2073-8994/2/4/1945
We argue that quantum theory should proceed not from a spacetime background but from a Lie algebra, which is treated as a symmetry algebra. Then the fact that the cosmological constant is positive means not that the spacetime background is curved but that the de Sitter (dS) algebra as the symmetry algebra is more relevant than the Poincare or anti de Sitter ones. The physical interpretation of irreducible representations (IRs) of the dS algebra is considerably different from that for the other two algebras. One IR of the dS algebra splits into independent IRs for a particle and its antiparticle only when Poincare approximation works with a high accuracy. Only in this case additive quantum numbers such as electric, baryon and lepton charges are conserved, while at early stages of the Universe they could not be conserved. Another property of IRs of the dS algebra is that only fermions can be elementary and there can be no neutral elementary particles. The cosmological repulsion is a simple kinematical consequence of dS symmetry on quantum level when quasiclassical approximation is valid. Therefore the cosmological constant problem does not exist and there is no need to involve dark energy or other fields for explaining this phenomenon (in agreement with a similar conclusion by Bianchi and Rovelli).Symmetry2010-11-1924Article10.3390/sym2041945194519802073-89942010-11-19doi: 10.3390/sym2041945Felix M. Lev<![CDATA[Symmetry, Vol. 2, Pages 1925-1944: Three-Dimensional Facial Asymmetry in Attractive and Normal People from Childhood to Young Adulthood]]>
http://www.mdpi.com/2073-8994/2/4/1925
We are currently investigating measurable esthetic characteristics in persons considered “attractive” by the media. Three-dimensional soft-tissue facial asymmetry was quantified in 380 attractive (148 males, 232 females) and 669 control (397 males, 272 females) healthy persons aged 4–30 years. The coordinates of 50 facial landmarks were collected by a computerized digitizer, and asymmetry computed. Soft-tissue facial asymmetries reduced as a function of age in all cases. Attractive children were more symmetric than control children, but the reverse was true for young adults. The effect of symmetry on attractiveness seems to change as a function of age.Symmetry2010-11-0924Article10.3390/sym2041925192519442073-89942010-11-09doi: 10.3390/sym2041925Chiarella SforzaAlberto LainoGaia GrandiLuca PisoniVirgilio Ferruccio Ferrario