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Symmetry, Volume 2, Issue 3 (September 2010), Pages 1201-1744

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Research

Jump to: Review

Open AccessArticle Will Science and Consciousness Ever Meat? Complexity, Symmetry and Qualia
Symmetry 2010, 2(3), 1250-1269; doi:10.3390/sym2031250
Received: 9 March 2010 / Revised: 25 May 2010 / Accepted: 23 June 2010 / Published: 25 June 2010
PDF Full-text (145 KB) | Supplementary Files
Abstract
Within recent discussions in the Philosophy of Mind, the nature of conscious phenomenal states or qualia (also called ‘raw feels’ or the feel of ‘what it is like to be’) has been an important focus of interest. Proponents of Mind-Body Type-Identity [...] Read more.
Within recent discussions in the Philosophy of Mind, the nature of conscious phenomenal states or qualia (also called ‘raw feels’ or the feel of ‘what it is like to be’) has been an important focus of interest. Proponents of Mind-Body Type-Identity theories have claimed that mental states can be reduced to neurophysiological states of the brain. Others have denied that such a reduction is possible; for them, there remains an explanatory gap. In this paper, functionalist, physicalist, epiphenomenalist, and biological models of the mind are discussed and compared. Donald Davidson’s Anomalous Monism is proposed as a unifying framework for a non-reductive theory of qualia and consciousness. Downward Causation, Emergence through Symmetry-breaking, and Dynamical Systems Theory are used to show how consciousness and qualia emerge from their neural substrate and can also be causally efficacious. Full article
(This article belongs to the Special Issue Complexity and Symmetry)
Open AccessArticle Emergent Dynamics of Five-Colour QCD Due to Dimensional Frustration
Symmetry 2010, 2(3), 1322-1337; doi:10.3390/sym2031322
Received: 24 November 2009 / Revised: 20 May 2010 / Accepted: 25 June 2010 / Published: 1 July 2010
PDF Full-text (261 KB)
Abstract
The consequences for five-colour QCD of a novel symmetry-breaking mechanism, published in an earlier paper, are further explored. In addition to the emergence of QED and three-colour QCD, there is also a candidate for the Z0μ. The representation theory [...] Read more.
The consequences for five-colour QCD of a novel symmetry-breaking mechanism, published in an earlier paper, are further explored. In addition to the emergence of QED and three-colour QCD, there is also a candidate for the Z0μ. The representation theory of SU (N) is applied to the matter sector and yields the quark and electron charge ratios, and a mechanism for generating fermion particle masses. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle How to Find the Fries Structures for Benzenoid Hydrocarbons
Symmetry 2010, 2(3), 1390-1400; doi:10.3390/sym2031390
Received: 3 February 2010 / Revised: 4 June 2010 / Accepted: 28 June 2010 / Published: 6 July 2010
Cited by 6 | PDF Full-text (286 KB)
Abstract
An efficient algorithm leading to the Fries canonical structure is presented for benzenoid hydrocarbons. This is a purely topological approach, which is based on adjacency matrices and the Hadamard procedure of matrix multiplication. The idea is presented for naphthalene, as an example. [...] Read more.
An efficient algorithm leading to the Fries canonical structure is presented for benzenoid hydrocarbons. This is a purely topological approach, which is based on adjacency matrices and the Hadamard procedure of matrix multiplication. The idea is presented for naphthalene, as an example. The Fries canonical-structures are also derived for anthracene, coronene, triphenylene, phenanthrene, benz[a]pyrene, and one large benzenoid system. The Fries concept can be convenient for obtaining Clar structures with the maximum number of sextets, which in turn effectively represent π-electron (de)localization in benzenoid hydrocarbons. Full article
(This article belongs to the Special Issue Aromaticity and Molecular Symmetry)
Open AccessArticle Symmetry, Symmetry Breaking and Topology
Symmetry 2010, 2(3), 1401-1422; doi:10.3390/sym2031401
Received: 1 February 2010 / Revised: 30 April 2010 / Accepted: 5 July 2010 / Published: 7 July 2010
Cited by 2 | PDF Full-text (172 KB)
Abstract
The ground state of a system with symmetry can be described by a group G. This symmetry group G can be discrete or continuous. Thus for a crystal G is a finite group while for the vacuum state of a grand [...] Read more.
The ground state of a system with symmetry can be described by a group G. This symmetry group G can be discrete or continuous. Thus for a crystal G is a finite group while for the vacuum state of a grand unified theory G is a continuous Lie group. The ground state symmetry described by G can change spontaneously from G to one of its subgroups H as the external parameters of the system are modified. Such a macroscopic change of the ground state symmetry of a system from G to H correspond to a “phase transition”. Such phase transitions have been extensively studied within a framework due to Landau. A vast range of systems can be described using Landau’s approach, however there are also systems where the framework does not work. Recently there has been growing interest in looking at such non-Landau type of phase transitions. For instance there are several “quantum phase transitions” that are not of the Landau type. In this short review we first describe a refined version of Landau’s approach in which topological ideas are used together with group theory. The combined use of group theory and topological arguments allows us to determine selection rule which forbid transitions from G to certain of its subgroups. We end by making a few brief remarks about non-Landau type of phase transition. Full article
(This article belongs to the Special Issue Symmetry Breaking Phenomena)
Open AccessArticle Orientational Sampling Schemes Based on Four Dimensional Polytopes
Symmetry 2010, 2(3), 1423-1449; doi:10.3390/sym2031423
Received: 4 February 2010 / Revised: 27 May 2010 / Accepted: 30 June 2010 / Published: 7 July 2010
Cited by 5 | PDF Full-text (654 KB)
Abstract
The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may [...] Read more.
The vertices of regular four-dimensional polytopes are used to generate sets of uniformly distributed three-dimensional rotations, which are provided as tables of Euler angles. The spherical moments of these orientational sampling schemes are treated using group theory. The orientational sampling sets may be used in the numerical computation of solid-state nuclear magnetic resonance spectra, and in spherical tensor analysis procedures. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessArticle SU(2) and SU(1,1) Approaches to Phase Operators and Temporally Stable Phase States: Applications to Mutually Unbiased Bases and Discrete Fourier Transforms
Symmetry 2010, 2(3), 1461-1484; doi:10.3390/sym2031461
Received: 9 June 2010 / Revised: 8 July 2010 / Accepted: 9 July 2010 / Published: 12 July 2010
Cited by 5 | PDF Full-text (225 KB)
Abstract
We propose a group-theoretical approach to the generalized oscillator algebra Aκ recently investigated in J. Phys. A: Math. Theor. 2010, 43, 115303. The case κ ≥ 0 corresponds to the noncompact group SU(1,1) (as for the harmonic oscillator and the Pöschl-Teller systems) while the case κ < 0 is described by the compact group SU(2) (as for the Morse system). We construct the phase operators and the corresponding temporally stable phase eigenstates for Aκ in this group-theoretical context. The SU(2) case is exploited for deriving families of mutually unbiased bases used in quantum information. Along this vein, we examine some characteristics of a quadratic discrete Fourier transform in connection with generalized quadratic Gauss sums and generalized Hadamard matrices. Full article
Open AccessArticle On the Harmonic Oscillator Model of Electron Delocalization (HOMED) Index and its Application to Heteroatomic π-Electron Systems
Symmetry 2010, 2(3), 1485-1509; doi:10.3390/sym2031485
Received: 31 December 2009 / Revised: 19 May 2010 / Accepted: 7 July 2010 / Published: 12 July 2010
Cited by 43 | PDF Full-text (260 KB)
Abstract
The HOMA (Harmonic Oscillator Model of Aromaticity) index, reformulated in 1993, has been very often applied to describe π-electron delocalization for mono- and polycyclic π-electron systems. However, different measures of π-electron delocalization were employed for the CC, CX, and XY bonds, and [...] Read more.
The HOMA (Harmonic Oscillator Model of Aromaticity) index, reformulated in 1993, has been very often applied to describe π-electron delocalization for mono- and polycyclic π-electron systems. However, different measures of π-electron delocalization were employed for the CC, CX, and XY bonds, and this index seems to be inappropriate for compounds containing heteroatoms. In order to describe properly various resonance effects (σ-π hyperconjugation, n-π conjugation, π-π conjugation, and aromaticity) possible for heteroatomic π-electron systems, some modifications, based on the original HOMA idea, were proposed and tested for simple DFT structures containing C, N, and O atoms. An abbreviation HOMED was used for the modified index. Full article
(This article belongs to the Special Issue Aromaticity and Molecular Symmetry)
Open AccessArticle Chemical Reasoning Based on an Invariance Property: Bond and Lone Pair Pictures in Quantum Structural Formulas
Symmetry 2010, 2(3), 1559-1590; doi:10.3390/sym2031559
Received: 21 December 2009 / Revised: 9 July 2010 / Accepted: 22 July 2010 / Published: 23 July 2010
PDF Full-text (2526 KB)
Abstract
Chemists use one set of orbitals when comparing to a structural formula, hybridized AOs or NBOs for example, and another for reasoning in terms of frontier orbitals, MOs usually. Chemical arguments can frequently be made in terms of energy and/or electron density [...] Read more.
Chemists use one set of orbitals when comparing to a structural formula, hybridized AOs or NBOs for example, and another for reasoning in terms of frontier orbitals, MOs usually. Chemical arguments can frequently be made in terms of energy and/or electron density without the consideration of orbitals at all. All orbital representations, orthogonal or not, within a given function space are related by linear transformation. Chemical arguments based on orbitals are really energy or electron density arguments; orbitals are linked to these observables through the use of operators. The Valency Interaction Formula, VIF, offers a system of chemical reasoning based on the invariance of observables from one orbital representation to another. VIF pictures have been defined as one-electron density and Hamiltonian operators. These pictures are classified in a chemically meaningful way by use of linear transformations applied to them in the form of two pictorial rules and the invariance of the number of doubly, singly, and unoccupied orbitals or bonding, nonbonding, and antibonding orbitals under these transformations. The compatibility of the VIF method with the bond pair – lone pair language of Lewis is demonstrated. Different electron lone pair representations are related by the pictorial rules and have stability understood in terms of Walsh’s rules. Symmetries of conjugated ring systems are related to their electronic state by simple mathematical formulas. Description of lone pairs in conjugated systems is based on the strength and sign of orbital interactions around the ring. Simple models for bonding in copper clusters are tested, and the bonding of O2 to Fe(II) in hemoglobin is described. Arguments made are supported by HF, B3LYP, and MP2 computations. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
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Open AccessArticle Asymmetry, Symmetry and Beauty
Symmetry 2010, 2(3), 1591-1624; doi:10.3390/sym2031591
Received: 2 June 2010 / Revised: 10 July 2010 / Accepted: 16 July 2010 / Published: 30 July 2010
Cited by 4 | PDF Full-text (2105 KB)
Abstract
Asymmetry and symmetry coexist in natural and human processes.  The vital role of symmetry in art has been well demonstrated. This article highlights the complementary role of asymmetry. Further we show that the interaction of asymmetric action (recursion) and symmetric opposition (sinusoidal [...] Read more.
Asymmetry and symmetry coexist in natural and human processes.  The vital role of symmetry in art has been well demonstrated. This article highlights the complementary role of asymmetry. Further we show that the interaction of asymmetric action (recursion) and symmetric opposition (sinusoidal waves) are instrumental in generating creative features (relatively low entropy, temporal complexity, novelty (less recurrence in the data than in randomized copies and complex frequency composition). These features define Bios, a pattern found in musical compositions and in poetry, except for recurrence instead of novelty. Bios is a common pattern in many natural and human processes (quantum processes, the expansion of the universe, gravitational waves, cosmic microwave background radiation, DNA, physiological processes, animal and human populations, and economic time series). The reduction in entropy is significant, as it reveals creativity and contradicts the standard claim of unavoidable decay towards disorder. Artistic creations capture fundamental features of the world. Full article
(This article belongs to the Special Issue Symmetry and Beauty)
Open AccessArticle Mirror Symmetry Breaking in Helical Polysilanes: Preference between Left and Right of Chemical and Physical Origin
Symmetry 2010, 2(3), 1625-1652; doi:10.3390/sym2031625
Received: 21 February 2010 / Revised: 2 August 2010 / Accepted: 9 August 2010 / Published: 13 August 2010
Cited by 12 | PDF Full-text (1009 KB) | Supplementary Files
Abstract
From elemental particles to human beings, matter is dissymmetric with respect to mirror symmetry. In 1860, Pasteur conjectured that biomolecular handedness— homochirality—may originate from certain inherent dissymmetric forces existing in the universe. Kipping, a pioneer of organosilicon chemistry, was interested in the [...] Read more.
From elemental particles to human beings, matter is dissymmetric with respect to mirror symmetry. In 1860, Pasteur conjectured that biomolecular handedness— homochirality—may originate from certain inherent dissymmetric forces existing in the universe. Kipping, a pioneer of organosilicon chemistry, was interested in the handedness of sodium chlorate during his early research life. Since Kipping first synthesized several Si-Si bonded oligomers bearing phenyl groups, Si-Si bonded high polymers carrying various organic groups—polysilanes—can be prepared by sodium-mediated condensation of the corresponding organodichlorosilanes. Among these polysilanes, optically active helical polysilanes with enantiomeric pairs of organic side groups may be used for testing the mirror symmetry-breaking hypothesis by weak neutral current (WNC) origin in the realm of chemistry and material science. Several theoretical studies have predicted that WNC-existing chiral molecules with stereogenic centers and/or stereogenic bonds allow for distinguishing between image and mirror image molecules. Based on several amplification mechanisms, theorists claimed that minute differences, though still very subtle, may be detectable by precise spectroscopic and physicochemical measurements if proper chiral molecular pairs were employed. The present paper reports comprehensively an inequality between six pairs of helical polysilane high polymers, presumably, detectable by (chir)optical and achiral 29Si-/13C- NMR spectra, and viscometric measurements. Full article
(This article belongs to the Special Issue Symmetry of Life and Homochirality)
Open AccessArticle On the Importance of Clar Structures of Polybenzenoid Hydrocarbons as Revealed by the π-Contribution to the Electron Localization Function
Symmetry 2010, 2(3), 1653-1682; doi:10.3390/sym2031653
Received: 29 June 2010 / Revised: 11 August 2010 / Accepted: 18 August 2010 / Published: 20 August 2010
Cited by 19 | PDF Full-text (988 KB)
Abstract
The degree of p-electron (de)localization and aromaticity of a series of polybenzenoid hydrocarbons (PBHs) has been analyzed through the π-contribution to the electron localization function (ELFπ), calculated at the B3LYP/6-311G(d,p) hybrid density functional theory level. The extent of p-electron delocalization [...] Read more.
The degree of p-electron (de)localization and aromaticity of a series of polybenzenoid hydrocarbons (PBHs) has been analyzed through the π-contribution to the electron localization function (ELFπ), calculated at the B3LYP/6-311G(d,p) hybrid density functional theory level. The extent of p-electron delocalization in the various hexagons of a PBH was determined through analysis of the bifurcation values of the ELFp basins (BV(ELFp)), the spans in the bifurcation values in each hexagon (ΔBV(ELFπ)), and the ring-closure bifurcation values of the ELFπ (RCBV(ELFπ)). These computed results were compared to the qualitative description of local aromaticities of the different hexagons in terms of Clar structures with p-sextets. Benzene, [18]annulene, and thirty two PBHs were analyzed at their equilibrium geometries, and benzene and triphenylene were also analyzed at bond length distorted structures. In general, the description of PBHs in terms of Clar valence structures is supported by the ELFp properties, although there are exceptions. For PBHs at their equilibrium geometries there is a clear sigmoidal relationship between the CC bond lengths and the amount of p-electron (de)localization at these bonds, however, this relationship is lost for bond distorted geometries. In the latter cases, we specifically examined benzene in D3h symmetric “1,3,5-cyclohexatriene” structures and triphenylene in eight different structures. From the distorted benzenes and triphenylenes it becomes clear that there is a distinct tendency for the p-electron network to retain delocalization (aromaticity). The ELFp analysis thus reveals an antidistortive rather than a distortive behavior of the p-electrons in these investigated compounds. Full article
(This article belongs to the Special Issue Aromaticity and Molecular Symmetry)
Open AccessArticle Complex Networks and Symmetry II: Reciprocity and Evolution of World Trade
Symmetry 2010, 2(3), 1710-1744; doi:10.3390/sym2031710
Received: 25 August 2010 / Revised: 17 September 2010 / Accepted: 20 September 2010 / Published: 27 September 2010
Cited by 7 | PDF Full-text (727 KB)
Abstract
We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade [...] Read more.
We exploit the symmetry concepts developed in the companion review of this article to introduce a stochastic version of link reversal symmetry, which leads to an improved understanding of the reciprocity of directed networks. We apply our formalism to the international trade network and show that a strong embedding in economic space determines particular symmetries of the network, while the observed evolution of reciprocity is consistent with a symmetry breaking taking place in production space. Our results show that networks can be strongly affected by symmetry-breaking phenomena occurring in embedding spaces, and that stochastic network symmetries can successfully suggest, or rule out, possible underlying mechanisms. Full article
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Review

Jump to: Research

Open AccessReview Loss of Temporal Homogeneity and Symmetry in Statistical Systems: Deterministic Versus Stochastic Dynamics
Symmetry 2010, 2(3), 1201-1249; doi:10.3390/sym2031201
Received: 13 January 2010 / Revised: 9 June 2010 / Accepted: 21 June 2010 / Published: 24 June 2010
Cited by 10 | PDF Full-text (596 KB)
Abstract
A detailed analysis of deterministic (one-to-one) and stochastic (one-to-many) dynamics establishes that dS/dt > 0 is only consistent with the latter, which contains violation of temporal symmetry and homogeneity. We observe that the former only supports dS/dt = 0 and cannot give [...] Read more.
A detailed analysis of deterministic (one-to-one) and stochastic (one-to-many) dynamics establishes that dS/dt > 0 is only consistent with the latter, which contains violation of temporal symmetry and homogeneity. We observe that the former only supports dS/dt = 0 and cannot give rise to Boltzmann’s molecular chaos assumption. The ensemble average is more meaningful than the temporal average, especially in non-equilibrium statistical mechanics of systems confined to disjoint phase space components, which commonly occurs at low temperatures. We propose that the stochasticity arises from extra degrees of freedom, which are not part of the system. We provide a simple resolution of the recurrence and irreversibility paradoxes. Full article
(This article belongs to the Special Issue Entropy, Order and Symmetry)
Open AccessReview Symmetry and Asymmetry in Bouncing Gaits
Symmetry 2010, 2(3), 1270-1321; doi:10.3390/sym2031270
Received: 16 December 2009 / Revised: 18 June 2010 / Accepted: 24 June 2010 / Published: 25 June 2010
Cited by 8 | PDF Full-text (1898 KB) | Supplementary Files
Abstract
In running, hopping and trotting gaits, the center of mass of the body oscillates each step below and above an equilibrium position where the vertical force on the ground equals body weight. In trotting and low speed human running, the average vertical [...] Read more.
In running, hopping and trotting gaits, the center of mass of the body oscillates each step below and above an equilibrium position where the vertical force on the ground equals body weight. In trotting and low speed human running, the average vertical acceleration of the center of mass during the lower part of the oscillation equals that of the upper part, the duration of the lower part equals that of the upper part and the step frequency equals the resonant frequency of the bouncing system: we define this as on-offground symmetric rebound. In hopping and high speed human running, the average vertical acceleration of the center of mass during the lower part of the oscillation exceeds that of the upper part, the duration of the upper part exceeds that of the lower part and the step frequency is lower than the resonant frequency of the bouncing system: we define this as on-off-ground asymmetric rebound. Here we examine the physical and physiological constraints resulting in this on-off-ground symmetry and asymmetry of the rebound. Furthermore, the average force exerted during the brake when the body decelerates downwards and forwards is greater than that exerted during the push when the body is reaccelerated upwards and forwards. This landing-takeoff asymmetry, which would be nil in the elastic rebound of the symmetric spring-mass model for running and hopping, suggests a less efficient elastic energy storage and recovery during the bouncing step. During hopping, running and trotting the landing-takeoff asymmetry and the mass-specific vertical stiffness are smaller in larger animals than in the smaller animals suggesting a more efficient rebound in larger animals. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
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Open AccessReview Phase Diagram and Critical Properties within an Effective Model of QCD: The Nambu–Jona-Lasinio Model Coupled to the Polyakov Loop
Symmetry 2010, 2(3), 1338-1374; doi:10.3390/sym2031338
Received: 19 March 2010 / Revised: 27 May 2010 / Accepted: 2 July 2010 / Published: 6 July 2010
Cited by 31 | PDF Full-text (890 KB)
Abstract
We investigate the phase diagram of the so-called Polyakov–Nambu–Jona-Lasinio model at finite temperature and non-zero chemical potential with three quark flavors. Chiral and deconfinement phase transitions are discussed and the relevant order-like parameters are analyzed. The results are compared with simple thermodynamic [...] Read more.
We investigate the phase diagram of the so-called Polyakov–Nambu–Jona-Lasinio model at finite temperature and non-zero chemical potential with three quark flavors. Chiral and deconfinement phase transitions are discussed and the relevant order-like parameters are analyzed. The results are compared with simple thermodynamic expectations and lattice data. We present the phase diagram in the (T, μB) plane, paying special attention to the critical end point: as the strength of the flavor-mixing interaction becomes weaker, the critical end point moves to low temperatures and can even disappear. Full article
(This article belongs to the Special Issue Symmetry Breaking Phenomena)
Open AccessReview Symmetric Matrix Fields in the Finite Element Method
Symmetry 2010, 2(3), 1375-1389; doi:10.3390/sym2031375
Received: 1 April 2010 / Revised: 3 June 2010 / Accepted: 30 June 2010 / Published: 6 July 2010
Cited by 4 | PDF Full-text (148 KB)
Abstract
The theory of elasticity is used to predict the response of a material body subject to applied forces. In the linear theory, where the displacement is small, the stress tensor which measures the internal forces is the variable of primal importance. However [...] Read more.
The theory of elasticity is used to predict the response of a material body subject to applied forces. In the linear theory, where the displacement is small, the stress tensor which measures the internal forces is the variable of primal importance. However the symmetry of the stress tensor which expresses the conservation of angular momentum had been a challenge for finite element computations. We review in this paper approaches based on mixed finite element methods. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
Open AccessReview Possible Physical Mechanisms in the Galaxy to Cause Homochiral Biomaterials for Life
Symmetry 2010, 2(3), 1450-1460; doi:10.3390/sym2031450
Received: 2 February 2010 / Revised: 8 May 2010 / Accepted: 21 June 2010 / Published: 9 July 2010
Cited by 1 | PDF Full-text (438 KB)
Abstract The origin of homochirality in life remains a mystery that some believe is essential for life, and which may result from chiral symmetry breaking interactions with galactic organic material. Full article
(This article belongs to the Special Issue Symmetry of Life and Homochirality)
Open AccessReview Behind the Looking-Glass: A Review on Human Symmetry Perception
Symmetry 2010, 2(3), 1510-1543; doi:10.3390/sym2031510
Received: 25 June 2010 / Revised: 13 July 2010 / Accepted: 19 July 2010 / Published: 22 July 2010
Cited by 38 | PDF Full-text (5565 KB)
Abstract
The human visual system is highly proficient in extracting bilateral symmetry from visual input. This paper reviews empirical and theoretical work on human symmetry perception with a focus on recent issues such as its neural underpinnings. Symmetry detection is shown to be [...] Read more.
The human visual system is highly proficient in extracting bilateral symmetry from visual input. This paper reviews empirical and theoretical work on human symmetry perception with a focus on recent issues such as its neural underpinnings. Symmetry detection is shown to be a versatile, ongoing visual process that interacts with other visual processes. Evidence seems to converge towards the idea that  symmetry detection is subserved by a preprocessing stage involving spatial filters followed by information integration across the visual field in higher-tier cortical areas. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
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Open AccessReview Symmetries of the Central Vestibular System: Forming Movements for Gravity and a Three-Dimensional World
Symmetry 2010, 2(3), 1544-1558; doi:10.3390/sym2031544
Received: 18 May 2010 / Accepted: 16 July 2010 / Published: 22 July 2010
Cited by 2 | PDF Full-text (403 KB)
Abstract
Intrinsic dynamics of the central vestibular system (CVS) appear to be at least partly determined by the symmetries of its connections. The CVS contributes to whole-body functions such as upright balance and maintenance of gaze direction. These functions coordinate disparate senses (visual, [...] Read more.
Intrinsic dynamics of the central vestibular system (CVS) appear to be at least partly determined by the symmetries of its connections. The CVS contributes to whole-body functions such as upright balance and maintenance of gaze direction. These functions coordinate disparate senses (visual, inertial, somatosensory, auditory) and body movements (leg, trunk, head/neck, eye). They are also unified by geometric conditions. Symmetry groups have been found to structure experimentally-recorded pathways of the central vestibular system. When related to geometric conditions in three-dimensional physical space, these symmetry groups make sense as a logical foundation for sensorimotor coordination. Full article
(This article belongs to the Special Issue Feature Papers: Symmetry Concepts and Applications)
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Open AccessReview Complex Networks and Symmetry I: A Review
Symmetry 2010, 2(3), 1683-1709; doi:10.3390/sym2031683
Received: 25 August 2010 / Revised: 17 September 2010 / Accepted: 20 September 2010 / Published: 27 September 2010
Cited by 12 | PDF Full-text (458 KB)
Abstract
In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks [...] Read more.
In this review we establish various connections between complex networks and symmetry. While special types of symmetries (e.g., automorphisms) are studied in detail within discrete mathematics for particular classes of deterministic graphs, the analysis of more general symmetries in real complex networks is far less developed. We argue that real networks, as any entity characterized by imperfections or errors, necessarily require a stochastic notion of invariance. We therefore propose a definition of stochastic symmetry based on graph ensembles and use it to review the main results of network theory from an unusual perspective. The results discussed here and in a companion paper show that stochastic symmetry highlights the most informative topological properties of real networks, even in noisy situations unaccessible to exact techniques. Full article
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