Research

Jump to: Review

1054 KiB

Open AccessArticle

Wigner’s Space-Time Symmetries Based on the Two-by-Two Matrices of the Damped Harmonic Oscillators and the Poincaré Sphere

by Sibel Başkal, Young S. Kim and Marilyn E. Noz

Symmetry 2014, 6(3), 473-515; https://doi.org/10.3390/sym6030473 - 25 Jun 2014

Cited by 6 | Viewed by 7990

Abstract

The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group Sp(2). It is shown that this oscillator system contains the essential features of Wigner’s little groups dictating the [...] Read more.

The second-order differential equation for a damped harmonic oscillator can be converted to two coupled first-order equations, with two two-by-two matrices leading to the group Sp(2). It is shown that this oscillator system contains the essential features of Wigner’s little groups dictating the internal space-time symmetries of particles in the Lorentz-covariant world. The little groups are the subgroups of the Lorentz group whose transformations leave the four-momentum of a given particle invariant. It is shown that the damping modes of the oscillator correspond to the little groups for massive and imaginary-mass particles respectively. When the system makes the transition from the oscillation to damping mode, it corresponds to the little group for massless particles. Rotations around the momentum leave the four-momentum invariant. This degree of freedom extends the Sp(2) symmetry to that of SL(2, c) corresponding to the Lorentz group applicable to the four-dimensional Minkowski space. The Poincaré sphere contains the SL(2, c) symmetry. In addition, it has a non-Lorentzian parameter allowing us to reduce the mass continuously to zero. It is thus possible to construct the little group for massless particles from that of the massive particle by reducing its mass to zero. Spin-1/2 particles and spin-1 particles are discussed in detail. Full article

(This article belongs to the Special Issue Physics based on Two-by-two Matrices)

► Show Figures

1266 KiB

Open AccessArticle

Symmetry Adapted Assur Decompositions

by Anthony Nixon, Bernd Schulze, Adnan Sljoka and Walter Whiteley

Symmetry 2014, 6(3), 516-550; https://doi.org/10.3390/sym6030516 - 27 Jun 2014

Cited by 5 | Viewed by 6913

Abstract

Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, [...] Read more.

Assur graphs are a tool originally developed by mechanical engineers to decompose mechanisms for simpler analysis and synthesis. Recent work has connected these graphs to strongly directed graphs and decompositions of the pinned rigidity matrix. Many mechanisms have initial configurations, which are symmetric, and other recent work has exploited the orbit matrix as a symmetry adapted form of the rigidity matrix. This paper explores how the decomposition and analysis of symmetric frameworks and their symmetric motions can be supported by the new symmetry adapted tools. Full article

(This article belongs to the Special Issue Rigidity and Symmetry)

► Show Figures

697 KiB

Open AccessArticle

Gestalt Algebra—A Proposal for the Formalization of Gestalt Perception and Rendering

by Eckart Michaelsen

Symmetry 2014, 6(3), 566-577; https://doi.org/10.3390/sym6030566 - 07 Jul 2014

Cited by 12 | Viewed by 7167

Abstract

Gestalt Algebra gives a formal structure suitable for describing complex patterns in the image plain. This can be useful for recognizing hidden structure in images. The work at hand refers to the laws of perceptual psychology. A manifold called the Gestalt Domain is [...] Read more.

Gestalt Algebra gives a formal structure suitable for describing complex patterns in the image plain. This can be useful for recognizing hidden structure in images. The work at hand refers to the laws of perceptual psychology. A manifold called the Gestalt Domain is defined. Next to the position in 2D it also contains an orientation and a scale component. Algebraic operations on it are given for mirror symmetry as well as organization into rows. Additionally the Gestalt Domain contains an assessment component, and all the meaning of the operations implementing the Gestalt-laws is realized in the functions giving this component. The operation for mirror symmetry is binary, combining two parts into one aggregate as usual in standard algebra. The operation for organization into rows, however, combines n parts into an aggregate, where n may well be more than two. This is algebra in its more general sense. For recognition, primitives are extracted from digital raster images by Lowe’s Scale Invariant Feature Transform (SIFT). Lowe’s key-point descriptors can also be utilized. Experiments are reported with a set of images put forth for the Computer Vision and Pattern Recognition Workshops (CVPR) 2013 symmetry contest. Full article

(This article belongs to the Special Issue Visual Symmetry)

► Show Figures

Graphical abstract

243 KiB

Open AccessArticle

Coset Group Construction of Multidimensional Number Systems

by Horia I. Petrache

Symmetry 2014, 6(3), 578-588; https://doi.org/10.3390/sym6030578 - 11 Jul 2014

Cited by 1 | Viewed by 6039

Abstract

Extensions of real numbers in more than two dimensions, in particular quaternions and octonions, are finding applications in physics due to the fact that they naturally capture symmetries of physical systems. However, in the conventional mathematical construction of complex and multicomplex numbers multiplication [...] Read more.

Extensions of real numbers in more than two dimensions, in particular quaternions and octonions, are finding applications in physics due to the fact that they naturally capture symmetries of physical systems. However, in the conventional mathematical construction of complex and multicomplex numbers multiplication rules are postulated instead of being derived from a general principle. A more transparent and systematic approach is proposed here based on the concept of coset product from group theory. It is shown that extensions of real numbers in two or more dimensions follow naturally from the closure property of finite coset groups adding insight into the utility of multidimensional number systems in describing symmetries in nature. Full article

866 KiB

Open AccessArticle

Non-Crystallographic Layer Lattice Restrictions in Order-Disorder (OD) Structures

by Berthold Stöger

Symmetry 2014, 6(3), 589-621; https://doi.org/10.3390/sym6030589 - 21 Jul 2014

Cited by 4 | Viewed by 5560

Abstract

Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting [...] Read more.

Symmetry operations of layers periodic in two dimensions restrict the geometry the lattice according to the five two-dimensional Bravais types of lattices. In order-disorder (OD) structures, the operations relating equivalent layers generally leave invariant only a sublattice of the layers. The thus resulting restrictions can be expressed in terms of linear relations of the a², b² and a · b scalar products of the lattice basis vectors with rational coefficients. To characterize OD families and to check their validity, these lattice restrictions are expressed in the bases of different layers and combined. For a more familiar notation, they can be expressed in terms of the lattice parameters a, b and . Alternatively, the description of the lattice restrictions may be simplified by using centered lattices. The representation of the lattice restrictions in terms of scalar products is dependent on the chosen basis. A basis-independent classification of the lattice restrictions is outlined. Full article

(This article belongs to the Special Issue Crystal Symmetry and Structure)

► Show Figures

1119 KiB

Open AccessArticle

Lubricating and Waxy Esters. VI. Effect of Symmetry about Ester on Crystallization of Linear Monoester Isomers

by Laziz Bouzidi, Shaojun Li and Suresh S. Narine

Symmetry 2014, 6(3), 655-676; https://doi.org/10.3390/sym6030655 - 07 Aug 2014

Cited by 9 | Viewed by 6475

Abstract

The crystal structure development of jojoba-like esters incorporating either 1-decenoic acid and/or 1-decenol, namely octadec-9-enyl dec-9-enoate (JLE-28₁), and its isomer dec-9-enyl oleate (JLE-28₂) was investigated to reveal the effect of symmetry about the ester group on crystallization of aliphatic [...] Read more.

The crystal structure development of jojoba-like esters incorporating either 1-decenoic acid and/or 1-decenol, namely octadec-9-enyl dec-9-enoate (JLE-28₁), and its isomer dec-9-enyl oleate (JLE-28₂) was investigated to reveal the effect of symmetry about the ester group on crystallization of aliphatic fatty monoesters. The phase transformation path was investigated with temperature-time resolved X-ray diffraction during stepped isothermal crystallization, and while cooling from the melt at a fixed rate. Startling differences in phase behavior were uncovered between the isomers. When stepped isothermals were used, selective extinctions occurred at a transition temperature for JLE-28₁ but not for JLE-28₂. The extinctions, which are due to dramatic changes in the electronic density of certain families of planes, indicate a phase transition attributed to a brusque rearrangement of the oxygen atoms in the crystal subcell. The phase transition did not occur when the JLEs were cooled continuously. The crucial role played by the position of the alkyl chain and its orientation relative to the easy rotation site of the C–O bond in the phase trajectories of the JLEs was particularly highlighted. Full article

(This article belongs to the Special Issue Chemical Applications of Symmetry)

► Show Figures

5515 KiB

Open AccessArticle

Recognition of Symmetric 3D Bodies

by Tomáš Suk and Jan Flusser

Symmetry 2014, 6(3), 722-757; https://doi.org/10.3390/sym6030722 - 01 Sep 2014

Cited by 6 | Viewed by 11748

Abstract

The paper deals with the recognition of symmetric three-dimensional (3D) bodies that can be rotated and translated. We provide a complete list of all existing combinations of rotation and reflection symmetries in 3D. We define 3D complex moments by means of spherical harmonics, [...] Read more.

The paper deals with the recognition of symmetric three-dimensional (3D) bodies that can be rotated and translated. We provide a complete list of all existing combinations of rotation and reflection symmetries in 3D. We define 3D complex moments by means of spherical harmonics, and the influence of individual symmetry groups on complex moment values is studied. Each particular symmetry pre-defines certain moment values. These moments can no longer differentiate between two objects of the same symmetry, which decreases the recognition power of the feature set. They should not be included when constructing the invariants. Translation and rotation invariants up to the fourth order are presented and their performance is studied on both artificial and real data. Full article

► Show Figures

Graphical abstract

961 KiB

Open AccessArticle

Twinning of Polymer Crystals Suppressed by Entropy

by Nikos Ch. Karayiannis, Katerina Foteinopoulou and Manuel Laso

Symmetry 2014, 6(3), 758-780; https://doi.org/10.3390/sym6030758 - 04 Sep 2014

Cited by 10 | Viewed by 6804

Abstract

We propose an entropic argument as partial explanation of the observed scarcity of twinned structures in crystalline samples of synthetic organic polymeric materials. Polymeric molecules possess a much larger number of conformational degrees of freedom than low molecular weight substances. The preferred conformations [...] Read more.

We propose an entropic argument as partial explanation of the observed scarcity of twinned structures in crystalline samples of synthetic organic polymeric materials. Polymeric molecules possess a much larger number of conformational degrees of freedom than low molecular weight substances. The preferred conformations of polymer chains in the bulk of a single crystal are often incompatible with the conformations imposed by the symmetry of a growth twin, both at the composition surfaces and in the twin axis. We calculate the differences in conformational entropy between chains in single crystals and chains in twinned crystals, and find that the reduction in chain conformational entropy in the twin is sufficient to make the single crystal the stable thermodynamic phase. The formation of cyclic twins in molecular dynamics simulations of chains of hard spheres must thus be attributed to kinetic factors. In more realistic polymers this entropic contribution to the free energy can be canceled or dominated by nonbonded and torsional energetics. Full article

(This article belongs to the Special Issue Crystal Symmetry and Structure)

► Show Figures

Review

Jump to: Research

1652 KiB

Open AccessReview

Autosolvation: Architecture and Selection of Chiral Conformers in Alkylcobalt Carbonyl Molecular Clocks

by Róbert Kurdi, Attila Táborosi, Claudia Zucchi and Gyula Pályi

Symmetry 2014, 6(3), 551-565; https://doi.org/10.3390/sym6030551 - 04 Jul 2014

Cited by 6 | Viewed by 6465

Abstract

Autosolvation is an important factor in stabilizing the architecture of medium complicated molecules. It is a kind of “supramolecular force” acting in intramolecular manner, consisting of orbital-orbital interactions between polar groups, separated by more than one covalent bonds within the same molecule. This [...] Read more.

Autosolvation is an important factor in stabilizing the architecture of medium complicated molecules. It is a kind of “supramolecular force” acting in intramolecular manner, consisting of orbital-orbital interactions between polar groups, separated by more than one covalent bonds within the same molecule. This effect facilitates also the development of chiral conformations. Two typical alkylcobalt carbonyl type molecules are discussed here as examples of autosolvating intramolecular interactions, leading to dramatic selection of chiral conformers and indicating also to the limits of the effect. The conformers stabilized by autosolvation and their interconversion are excellent examples of a “molecular clockwork”. Operation mode of these molecular clockworks gives some insight into the intramolecular transfer of chiral information. Full article

(This article belongs to the Special Issue Supramolecular Chirality)

► Show Figures

Graphical abstract

20368 KiB

Open AccessReview

Symmetry Breaking in NMR Spectroscopy: The Elucidation of Hidden Molecular Rearrangement Processes

by Michael J. McGlinchey

Symmetry 2014, 6(3), 622-654; https://doi.org/10.3390/sym6030622 - 04 Aug 2014

Cited by 10 | Viewed by 13252

Abstract

Variable-temperature NMR spectroscopy is probably the most convenient and sensitive technique to monitor changes in molecular structure in solution. Rearrangements that are rapid on the NMR time-scale exhibit simplified spectra, whereby non-equivalent nuclear environments yield time-averaged resonances. At lower temperatures, when the rate [...] Read more.

Variable-temperature NMR spectroscopy is probably the most convenient and sensitive technique to monitor changes in molecular structure in solution. Rearrangements that are rapid on the NMR time-scale exhibit simplified spectra, whereby non-equivalent nuclear environments yield time-averaged resonances. At lower temperatures, when the rate of exchange is sufficiently reduced, these degeneracies are split and the underlying “static” molecular symmetry, as seen by X-ray crystallography, becomes apparent. Frequently, however, such rearrangement processes are hidden, even when they become slow on the NMR time-scale, because the molecular point group remains unchanged. Judicious symmetry breaking, such as by substitution of a molecular fragment by a similar, but not identical moiety, or by the incorporation of potentially diastereotopic (chemically non-equivalent) nuclei, allows the elucidation of the kinetics and energetics of such processes. Examples are chosen that include a wide range of rotations, migrations and other rearrangements in organic, inorganic and organometallic chemistry. Full article

(This article belongs to the Special Issue Chemical Applications of Symmetry)

► Show Figures

Graphical abstract

4220 KiB

Open AccessReview

Supramolecular Chirality: Solvent Chirality Transfer in Molecular Chemistry and Polymer Chemistry

by Michiya Fujiki

Symmetry 2014, 6(3), 677-703; https://doi.org/10.3390/sym6030677 - 13 Aug 2014

Cited by 74 | Viewed by 15799

Abstract

Controlled mirror symmetry breaking arising from chemical and physical origin is currently one of the hottest issues in the field of supramolecular chirality. The dynamic twisting abilities of solvent molecules are often ignored and unknown, although the targeted molecules and polymers in a [...] Read more.

Controlled mirror symmetry breaking arising from chemical and physical origin is currently one of the hottest issues in the field of supramolecular chirality. The dynamic twisting abilities of solvent molecules are often ignored and unknown, although the targeted molecules and polymers in a fluid solution are surrounded by solvent molecules. We should pay more attention to the facts that mostly all of the chemical and physical properties of these molecules and polymers in the ground and photoexcited states are significantly influenced by the surrounding solvent molecules with much conformational freedom through non-covalent supramolecular interactions between these substances and solvent molecules. This review highlights a series of studies that include: (i) historical background, covering chiral NaClO₃ crystallization in the presence of d-sugars in the late 19th century; (ii) early solvent chirality effects for optically inactive chromophores/fluorophores in the 1960s–1980s; and (iii) the recent development of mirror symmetry breaking from the corresponding achiral or optically inactive molecules and polymers with the help of molecular chirality as the solvent use quantity. Full article

(This article belongs to the Special Issue Supramolecular Chirality)

► Show Figures

Graphical abstract

13555 KiB

Open AccessReview

Adhesive/Repulsive Codes in Vertebrate Forebrain Morphogenesis

by Florencia Cavodeassi

Symmetry 2014, 6(3), 704-721; https://doi.org/10.3390/sym6030704 - 14 Aug 2014

Viewed by 6054

Abstract

The last fifteen years have seen the identification of some of the mechanisms involved in anterior neural plate specification, patterning, and morphogenesis, which constitute the first stages in the formation of the forebrain. These studies have provided us with a glimpse into the [...] Read more.

The last fifteen years have seen the identification of some of the mechanisms involved in anterior neural plate specification, patterning, and morphogenesis, which constitute the first stages in the formation of the forebrain. These studies have provided us with a glimpse into the molecular mechanisms that drive the development of an embryonic structure, and have resulted in the realization that cell segregation in the anterior neural plate is essential for the accurate progression of forebrain morphogenesis. This review summarizes the latest advances in our understanding of mechanisms of cell segregation during forebrain development, with and emphasis on the impact of this process on the morphogenesis of one of the anterior neural plate derivatives, the eyes. Full article

(This article belongs to the Special Issue Molecular Adhesion Codes in Geometric Regulation of Bilaterian Morphogenesis)

► Show Figures

23001 KiB

Open AccessReview

Chlorophylls, Symmetry, Chirality, and Photosynthesis

by Mathias O. Senge, Aoife A. Ryan, Kristie A. Letchford, Stuart A. MacGowan and Tamara Mielke

Symmetry 2014, 6(3), 781-843; https://doi.org/10.3390/sym6030781 - 10 Sep 2014

Cited by 169 | Viewed by 34902

Abstract

Chlorophylls are a fundamental class of tetrapyrroles and function as the central reaction center, accessory and photoprotective pigments in photosynthesis. Their unique individual photochemical properties are a consequence of the tetrapyrrole macrocycle, the structural chemistry and coordination behavior of the phytochlorin system, and [...] Read more.

Chlorophylls are a fundamental class of tetrapyrroles and function as the central reaction center, accessory and photoprotective pigments in photosynthesis. Their unique individual photochemical properties are a consequence of the tetrapyrrole macrocycle, the structural chemistry and coordination behavior of the phytochlorin system, and specific substituent pattern. They achieve their full potential in solar energy conversion by working in concert in highly complex, supramolecular structures such as the reaction centers and light-harvesting complexes of photobiology. The biochemical function of these structures depends on the controlled interplay of structural and functional principles of the apoprotein and pigment cofactors. Chlorophylls and bacteriochlorophylls are optically active molecules with several chiral centers, which are necessary for their natural biological function and the assembly of their supramolecular complexes. However, in many cases the exact role of chromophore stereochemistry in the biological context is unknown. This review gives an overview of chlorophyll research in terms of basic function, biosynthesis and their functional and structural role in photosynthesis. It highlights aspects of chirality and symmetry of chlorophylls to elicit further interest in their role in nature. Full article

(This article belongs to the Special Issue Supramolecular Chirality)

► Show Figures

Graphical abstract

Journal Menu

Journal Browser

Symmetry, Volume 6, Issue 3 (September 2014) – 13 articles , Pages 473-843

Research

Review

Further Information

Guidelines

MDPI Initiatives

Follow MDPI