Showing papers on "Symmetry (geometry) published in 1998"

Continuous Symmetry Measures. 5. The Classical Polyhedra.

Abstract: The continuous symmetry measures approach, designed to assess quantitatively the degree of any symmetry within any structure, is extended to the important class of the polyhedra. For this purpose, we developed a general methodology and a general computational tool, which identify the minimal distance of a given structure to a desired general shape with the same number of vertexes. Specifically, we employ this tool to evaluate quantitatively the degree of polyhedricity within distorted polyhedra, taking as examples the most central and abundant polyhedral structures in chemistry in general and in coordination chemistry in particular, namely the tetrahedron, the bipyramid, the octahedron, the cube, the icosahedron, and the dodecahedron. After describing the properties of the symmetry measurement tool, we show its application and versatility in a number of cases where the deviation from exact symmetry has been an issue, including z-axis Jahn−Teller type polyhedral distortions, tantalum hydride complexes, pen...

Facial symmetry and the perception of Beauty

Abstract: Evolutionary, as well as cultural, pressures may contribute to our perceptions of facial attractiveness. Biologists predict that facial symmetry should be attractive, because it may signal mate quality. We tested the prediction that facial symmetry is attractive by manipulating the symmetry of individual faces and observing the effect on attractiveness, and by examining whether natural variations in symmetry (between faces) correlated with perceived attractiveness. Attractiveness increased when we increased symmetry, and decreased when we reduced symmetry, in individual faces (Experiment 1), and natural variations in symmetry correlated significantly with attractiveness (Experiments 1 and 1A). Perfectly symmetric versions, made by blending the normal and mirror images of each face, were preferred to less symmetric versions of the same faces (even when those versions were also blends) (Experiments 1 and 2). Similar results were found when subjects judged the faces on appeal as a potential life partner, suggesting that facial symmetry may affect human mate choice. We conclude that facial symmetry is attractive and discuss the possibility that this preference for symmetry may be biologically based.

C3 Symmetry in Asymmetric Catalysis and Chiral Recognition.

Abstract: Rotational symmetry can be an important factor in the design of highly selective receptors for chiral recognition. This is well known for C2 -symmetric compounds, but the concept can be extended to chiral compounds of higher symmetry such as 1.

Detecting Symmetry in Grey Level Images: The Global Optimization Approach

Abstract: The detection of significant local reflectional symmetry in grey level images is considered. Prior segmentation is not assumed, and it is intended that the results could be used for guiding visual attention and for providing side information to segmentation algorithms. A local measure of reflectional symmetry that transforms the symmetry detection problem to a global optimization problem is defined. Reflectional symmetry detection becomes equivalent to finding the global maximum of a complicated multimodal function parameterized by the location of the center of the supporting region, its size, and the orientation of the symmetry axis. Unlike previous approaches, time consuming exhaustive search is avoided. A global optimization algorithm for solving the problem is presented. It is related to genetic algorithms and to adaptive random search techniques. The efficiency of the suggested algorithm is experimentally demonstrated. Just one thousand evaluations of the local symmetry measure are typically needed in order to locate the dominant symmetry in natural test images.

Classical 6j-symbols and the tetrahedron

Abstract: A classical 6j-symbol is a real number which can be associated to a labelling of the six edges of a tetrahedron by irreducible representations of SU(2). This abstract association is traditionally used simply to express the symmetry of the 6j-symbol, which is a purely algebraic object; however, it has a deeper geometric significance. Ponzano and Regge, expanding on work of Wigner, gave a striking (but unproved) asymptotic formula relating the value of the 6j-symbol, when the dimensions of the representations are large, to the volume of an honest Euclidean tetrahedron whose edge lengths are these dimensions. The goal of this paper is to prove and explain this formula by using geometric quantization. A surprising spin-off is that a generic Euclidean tetrahedron gives rise to a family of twelve scissors-congruent but non-congruent tetrahedra.

Equazioni a derivate parziali. - Monotonicity and symmetry of solutions of p-Laplace equations, 1 < p < 2, via the moving plane method

Abstract: In this paper we prove some monotonicity and symmetry properties of positive solutions of the equation - div Du) = f (u) satisfying an homogenuous Dirichlet boundary condition in a bounded domain Q. We assume 1 < p < 2 and f locally Lipschitz continuous and we do not require any hypothesis on the critical set of the solution. In particular we get that if Q is a ball then the solutions are radially symmetric and strictly radially decreasing.

SYMMOL: a program to find the maximum symmetry group in an atom cluster, given a prefixed tolerance

Abstract: SYMMOL is a new stand-alone program to find the maximum symmetry compatible with a given tolerance. The input requirements are cell parameters, atomic coordinates and the tolerance. The output is a set of symmetrized coordinates, the point-group label and the point-group elements. Some parameters quantifying the deviation from the exact symmetry are also calculated.

The Language of Mathematics: Making the Invisible Visible

Abstract: Counting reasoning and communication motion and change shape symmetry and regularity position chance the universe.

Chaotic Attractors with Discrete Planar Symmetries

Abstract: Chaotic behavior is known to be compatible with symmetry and illustrations are constructed using functions equivariant with respect to the desired symmetries. Earlier investigations determined families of equivariant functions for a few of the discrete symmetry groups in the plane; those results are extended to all the discrete symmetry groups of the plane. This includes consideration of the all the frieze and two-dimensional crystallographic groups. © 1998 Elsevier Science Ltd. All rights reserved.

Breaking the reflectional symmetry of interlimb coordination dynamics

Abstract: Interlimb rhythmic coordination is reflectionally symmetric when the left and right limb segments are identical in uncoupled frequencies and spatial orientation. In the present studies (4 experiments, with a total of 31 participants), when reflectional symmetry was broken through differences in timing (frequency), the resulting stable states were related by reflection and were identical for paired identically oriented limb segments behaving either as inverted or as ordinary pendulums. When reflectional symmetry was broken both temporally and spatially (coordinating inverted and ordinary pendular motions), the resulting stable states were different from those produced by identically oriented pendulums but nevertheless were related by reflection. In the Discussion, the authors focus on (a) symmetry breaking as leading to one of a number of symmetrically related states and (b) extending coordination dynamics with reflectional symmetry so that temporal and spatial asymmetries can both be accommodated.

On the Prediction of Crystal Morphology. II. Symmetry Roughening of Pairs of Connected Nets

Abstract: Within the framework of the periodic bond chain analysis for the prediction of crystal morphology, connected nets play a crucial role. For a face (hkl) often more than one connected net is found. Symmetry relations between such connected nets can give rise to symmetry roughening of nets. In this paper, all cases where symmetry may lead to symmetry roughening are derived. The role of a mother phase in contact with the crystal is explicitly taken into account. It turns out that cases of symmetry roughening are, in a sense, complementary to situations where the classical Bravais–Friedel–Donnay–Harker selection rules apply.

Two-dimensional orthogonal filter banks and wavelets with linear phase

Abstract: Two-dimensional (2-D) compactly supported, orthogonal wavelets and filter banks having linear phase are presented. Two cases are discussed: wavelets with two-fold symmetry (centrosymmetric) and wavelets with four-fold symmetry that are symmetric (or anti-symmetric) about the vertical and horizontal axes. We show that imposing the requirement of linear phase in the case of order-factorable wavelets imposes a simple constraint on each of its polynomial order-1 factors. We thus obtain a simple and complete method of constructing orthogonal order-factorable wavelets with linear phase. This method is exemplified by design in the case of four-band separable sampling. An interesting result that is similar to the one well-known in the one-dimensional (1-D) case is obtained: orthogonal order-factorable wavelets cannot be both continuous and have four-fold symmetry.

U-duality and M-theory, an algebraic approach

Abstract: Based on our work hep-th/9809039, we discuss how U-duality arises as an exact symmetry of M-theory from T-duality and 11D diffeomorphism invariance. A set of Weyl generators are shown to realize the Weyl group of SO(d,d,Z) and E_{d(d)}(Z), while Borel generators extend these finite groups into the full T- and U-duality groups. We discuss how the BPS states fall into various representations, and obtain duality invariant mass formulae, relevant for the computation of exact string amplitudes. The realization of U-duality symmetry in Matrix gauge theory is also considered.

S_4-symmetry of 6j-symbols and Frobenius-Schur indicators in rigid monoidal C^*-categories

Abstract: We show that a left-rigid monoidal C^*-category with irreducible monoidal unit is also a sovereign and spherical category. Defining a Frobenius-Schur type indicator we obtain selection rules for the fusion coefficients of irreducible objects. As a main result we prove S_4-invariance of 6j-symbols in such a category.

Segmenting by seeking the symmetry axis

Abstract: We introduce a method for segmenting a shape from an image and simultaneously determining its symmetry axis. The symmetry is used to help the segmentation and in turn the segmentation determines the symmetry. The problem is formulated as one of minimizing a goodness of fitness function and Dijkstra's algorithm is used to find the global minimum of the cost function. The results are illustrated on real images.

Effects of Pattern Orientation and Number of Symmetry Axes on the Detection of Mirror Symmetry in Dot and Solid Patterns

Abstract: It has been postulated that as the number of axes of symmetry in a pattern increases, so pattern ‘goodness’ increases. Recently, a distinction was made between two different theoretical accounts of regularity or ‘goodness’ in relation to patterns with mirror symmetry: the ‘transformational’ and the ‘holographic’ models. It was argued that the former predicts a ‘goodness’ ordering of four > three > two > one whereas the latter predicts four > two > three > one, where ‘>’ means greater regularity or goodness. In three experiments, we have tested these predictions. In experiment 1, we measured percentage correct and reaction time to dot patterns which had one, two, three, or four axes of symmetry and were flashed for 150 ms. Experiment 2 was identical except that patterns were presented for 2000 ms. In experiment 3, dot patterns were replaced by solid shapes which also had one, two, three, or four axes of symmetry. Although it was found that stimuli with four axes clearly allowed superior performance to that...

Evaluation method for optical system

Abstract: PROBLEM TO BE SOLVED: To evaluate imaging performance based on separation components by separating wavefront aberration having passed an optical system into a rotational symmetry component around an iris of the optical system and odd/even number symmetry components. SOLUTION: A light flux emitted from a light source 11 emitting light with a specific wavelength below 400 nm, for example, 248.4 nm passes through an illumination optical system 12 and illuminates a pattern on a mask 13 with a uniform illuminance. The light flux having passed the pattern forms a pattern image on a photosensitive surface on a wafer 15 via a projecting optical system 14 so that the pattern image is transferred on the photosensitive surface. For evaluating the imaging performance of the projecting optical system 14, the wavefront aberration on the imaging wavefront having passed the projection optical system 14 is firstly measured. Then, the wavefront aberration obtained by defining the coordinate system on the emission iris plane of the projection optical system 14 is expressed on the coordinates and expanded to orthogonal functions. For separating the wavefront aberration into the rotational symmetry component around the iris of the optical system and the odd number symmetry component and even number symmetry component, a polar coordinate is used and Zernike's cylindrical functions is also used as an orthogonal function.

Completely integrable systems with a symmetry in coordinates

Abstract: We explicitly construct the integrals of completely integrable quantum or classical systems whose potential functions are invariant under the action of a classical Weyl group. Our potential functions and integrals are expressed by the Weierstrass elliptic function.

Symmetry-Mode Analysis of Displacive Phase Transitions using International Tables for Crystallography

Abstract: The symmetry-mode analysis of a structural phase transition involves the use of methods of the representation theory of space groups. The aim of the paper is to present an alternative approach for the determination of the primary and secondary symmetry modes that contribute to the structural distortion of the low-symmetry phase. It only requires the systematic use of the data from International Tables for Crystallography, Vol. A, and is based on a fact, well known to crystallographers: the fully symmetrical displacements of any orbit of atoms (i.e. the modes compatible with the symmetry of the structure) follow in a straightforward way from the coordinate triplets of the corresponding Wyckoff positions. This property, systematically used for all intermediate subgroups between the space groups of the two phases, allows the determination of the relevant symmetry modes. Their distinction into primary and secondary modes comes out directly in the process of calculation. As an example, primary and secondary modes in the ferroelectric distortion of potassium selenate are divided.

Strong Symmetry Defined by Twisting Modules, Applied to Quasi-Hereditary Algebras with Triangular Decomposition and Vanishing Radical Cube

Abstract: For a finite dimensional algebra with triangular decomposition, a new kind of modules is defined, the twisting modules. Using the structure of these modules, for some algebras with vanishing radical cube the characteristic tilting module is described and its endomorphism ring is computed. This covers both Temperley–Lieb algebras and q-Schur algebras of finite representation type. Our basic tool is a new symmetry condition, stronger than the symmetry provided by the existence of a triangular decomposition in general.

Method of determining a symmetry line in a radiation image

Abstract: Method of determining (a) symmetry line(s) in a radiation image. A method of determining (a) symmetry line(s) in horizontal (vertical) direction in an image that is represented by a matrix of digital pixel values by (i) calculating a symmetry point for each column (row) of the image, and (ii) interconnecting calculated symmetry points.

Some remarks on the method of moving planes

Abstract: We propose a variational approach to the method of moving planes which easily applies to quasilinear equations of type (1-1) with $f$ locally Lipschitz continuous. To do this we use a characterization of Lipschitz continuous functions which allows us to get symmetry results without writing an equation for the difference between the solution and its reflection.