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Showing papers on "Symmetry (geometry) published in 1986"


Journal ArticleDOI
TL;DR: The symmetry of quasi-crystals, a class of materials that has recently aroused interest, is discussed in this article, where it is shown that a quasicrystal is a special case of an incommensurate crystal phase and that it can be described by a space group in more than three dimensions.
Abstract: The symmetry of quasi-crystals, a class of materials that has recently aroused interest, is discussed. It is shown that a quasi-crystal is a special case of an incommensurate crystal phase and that it can be described by a space group in more than three dimensions. A number of relevant three-dimensional quasi-crystals is discussed, in particular dihedral and icosahedral structures. The symmetry considerations are also applied to the two-dimensional Penrose patterns.

201 citations


Book ChapterDOI
TL;DR: In this article, the theory of quasiperiodic tilings and patterns is discussed. But this lecture deals with the theory only with quasi-periodic patterns, and not with quasiprocessor tilings.
Abstract: This lecture deals with the theory of quasiperiodic tilings, and more generally with quasiperiodic patterns. We present the ideas introduced in [9] and developed in [10].

115 citations


Journal ArticleDOI
TL;DR: In this paper, the S-matrix associated with a class of solvable potentials with SU(1, 1) dynamic symmetry can be computed in purely algebraic terms.

94 citations


Journal ArticleDOI
TL;DR: An asymptotically optimal algorithm to locate all the axes of mirror symmetry of a planar point set is presented by reducing the 2-D symmetry problem to linear pattern-matching.

62 citations



01 Jan 1986
TL;DR: In this article, the authors give an explicit description of the semigroup of values of a plane curve singularity with several branches in terms of the usual invariants of the equisingularity type in the sense of Zariski.
Abstract: We give an explicit description of the semigroup of values of a plane curve singularity with several branches in terms of the usual invariants of the equisingularity type in the sense of Zariski. The main tool is the set of elements called maximals, specially the absolute and the relative ones. First, we describe the semigroup in terms of the relative maximals and these ones in terms of the absolute maximals by means of a symmetry property which generalizes the well known property of symmetry for the singularities with only one branch. Then the absolute maximals are described in terms of the theory of maximal contact of higher genus developed by Lejeune.

57 citations


Journal ArticleDOI
TL;DR: In this paper, the authors present explicit results for aberration order nine for a spherical surface with symmetry algebra, which allows recursive computation of its aberration coefficients to arbitrarily high order.

28 citations


Journal ArticleDOI
TL;DR: In this paper, the authors studied the unfoldings of 0(2)-equivariant vector fields whose linearization has two pairs of purely imaginary eigenvalues, and they used isotropy subgroup techniques to classify the types of solutions which occur.
Abstract: In this paper we study the unfoldings of 0(2)-equivariant vector fields whose linearization has two pairs of purely imaginary eigenvalues. Such singularities may be expected to occur at isolated points in a centre manifold reduction of two-parameter systems with full circular symmetry. This situation differs from the corresponding non-symmetric system in that generically the eigenvalues may be either simple or double. The case when both eigenvalues are simple is similar to the Takens codimension-two singularity. Our interest lies in the cases where one or both of the purely imaginary eigenvalues are double; these cases lead to 6- and 8-dimensional centre manifolds, respectively. We use isotropy subgroup techniques to classify the types of solutions which occur. These include periodic solutions and 2-, 3-, and 4-dimensional invariant tori.

28 citations


Journal ArticleDOI
TL;DR: For the potential V (x ) = g 1 x 2 + g 2 x 4 + g 3 x 6 and its polynomial generalizations, the true and false energies of Killingbeck may both be interpreted as physical, corresponding to the two different forms of the potential.

27 citations


Journal ArticleDOI
TL;DR: From coding theory, the symmetry laws of aand P-tubulin subunits lead to the conclusion that 13 protofilaments passes one of the best known binary error-correcting codes [K, (13, 26, S)] with 64 code words.
Abstract: Oh (6/4) symmetry group describes face-centered-cubic sphere packing and derives informatkn coding laws.3 Hexagonal packing of protein monomers independent of the Oh (6/4) symmetry group has been used to explain the form patterns of viruses, flagella, and MT! Because hexagonal packing and face-centered-cubic packing of spheres have equal density, I use both to explain M T organization. Hexagonal packing may be used by fixing conditions. This is possible if tke centers of spheres lie on the surface of a cylinder (with radius equal to the Oh (6/4) unit sphere) and if (and only if) the sphere values in the axial direction (lattice) of the cylinder by order of sphere packing is the same as in the dimension in which the face-centered-cubic packing is done. Because 6-fold symmetry axis of Oh group is inverse, there must be two kinds of spheres (white and black) on the cylinder surface, but linked such that they have the dimension value in which the face-centered-cubic packing is done. This packing on the cylinder surface leads to “screw symmetry” (FIGURE 1). From coding theory, the symmetry laws of aand P-tubulin subunits lead to the conclusion that 13 protofilaments passes one of the best known binary error-correcting codes [K, (13, 26, S)] with 64 code words. Symmetry theory further suggests that on the surface of a circular cylinder in axial direction there must be a code of length of 24 monomer subunits. If the coding efficiency is used as a criterion of transmission, then 6-binary dimers of K, code should be coded to give a 4-dimer ternary sequence of Kz [24, 34, 131 code. This code may result from interaction between 24 tubulin monomers and high molecular weight proteins. Under the influence of Ca2+-calmodulin, binary dimers of K1 code give dimer ternary sequence of-K, code. In this way, KI and K2 codes, which result from the property of the Oh (6/4) symmetry group, lead a K (B6T4) transmission bioinforma-

22 citations


Journal ArticleDOI
TL;DR: In this article, the NMC principle is extended to non-totally symmetric irreducible representations of the point group to enable the selection rules for the physical properties from the symmetry of the object in question.
Abstract: The Neumann-Minnigerode-Curie Principle (NMC Principle) which enables one to derive the selection rules for the physical properties from the symmetry of the object in question is given for the totally symmetric representation of the point group and thus for the static properties of the object. Its dynamical properties, however, are described by the non-totally symmetric irreducible representations of the point group to which the NMC Principle can be extended.

Book ChapterDOI
01 Jan 1986
TL;DR: A Lie group is a "group" which is also a "manifold" as mentioned in this paper, which is the fundamental object in the field of differential geometry, generalizing the familiar concepts of curves and surfaces in 3D space.
Abstract: Roughly speaking, a Lie group is a “group” which is also a “manifold”. Of course, to make sense of this definition, we must explain these two basic concepts and how they can be related. Groups arise as an algebraic abstraction of the notion of symmetry; an important example is the group of rotations in the plane or three-dimensional space. Manifolds, which form the fundamental objects in the field of differential geometry, generalize the familiar concepts of curves and surfaces in three-dimensional space. In general, a manifold is a space that locally looks like Euclidean space, but whose global character might be quite different. The conjunction of these two seemingly disparate mathematical ideas combines, and significantly extends, both the algebraic methods of group theory and the multi-variable calculus used in analytic geometry. This resulting theory, particularly the powerful infinitesimal techniques, can then be applied to a wide range of physical and mathematical problems.

Journal ArticleDOI
TL;DR: In this article, it was shown that the only obstructions to global symmetry are presented by compositional factors, and that removing these factors allows us to straighten out complete level curves and to exploit the perfect symmetry of the functions remaining.
Abstract: A level curve for a nonconstant analytic function is a curve on which the function has constant modulus. Analytic functions should be symmetric about their level curves; any behavior on one side should be the reflection of behavior on the other. This symmetry may be realized locally-one straightens out a bit of the level curve with a local change of coordinates and appeals to the Schwarz reflection principle. Here we prove that, in fact, the only obstructions to global symmetry are presented by compositional factors. Removing these factors allows us to straighten out complete level curves and to exploit the perfect symmetry of the functions remaining. We can then establish, in a unified way, numerous old and new results about shared level curves and tracts-including work going back to Valiron and Cartwright and more recent work of Fuchs, Heins, and others. Given a Riemann surface O' and a nontrivial arc r in 4, we denote by Yr the nonconstant meromorphic functions having constant modulus one on F. When 4" is the Riemann sphere P, the plane C, or the unit disc D and F lies in the extended real line RX, we use the notation 3VR. The key theorem of our work is this:

Journal ArticleDOI
TL;DR: In this article, the authors describe general features and analyses a number of examples of various categories of colored and generalized symmetries used in visual arts, including Mexican, Egyptian, Celtic, New Guinean, Australian Aboriginal and Moorish, as well as 20th century ornamental art.
Abstract: The paper describes general features and analyses a number of examples of various categories of colored and generalized symmetries used in visual arts. After a thorough discussion of the problems of symmetry analysis and construction of dichroic and colored patterns, examples of South American. Mexican, Egyptian, Celtic, New Guinean, Australian Aboriginal and Moorish, as well as 20th century ornamental art are analysed. It is shown that with the exception of the modern creations by Hinterreiter and Escher, colored symmetry has been used much less than dichroic symmetry and has remained largely limited to unidirectional color modulation of uncolored or originally dichroic patterns. From among the categories of generalized symmetry, similarity, affine- and perspective-projective transformations are discussed and illustrated as well as catamorphy and the plane groups in non-Euclidean planes. Finally, the notions of submotif, submesh, supermesh, twinning, pseudosymmetry, noncommensurability of patterns, positional and “occupational” order-disorder phenomena, local high symmetry, dissymetry, enhancement of symmetry, dichotomy, modulation, antiphase boundaries and the hierarchy and superposition of symmetries are applied to the world of visual arts.

Journal ArticleDOI
TL;DR: In this article, the application of a mathematical principle, symmetry, to an area so highly variable as human culture seems remarkable until one discovers that the principle uncovers hitherto unknown consistencies in human behavior.
Abstract: The application of a mathematical principle, symmetry, to an area so highly variable as human culture seems remarkable until one discovers that the principle uncovers hitherto unknown consistencies in human behavior. In Sec. 2, the types of analyses which have shown how symmetry is manifested in culture are discussed. However, not all patterns created by cultural groups maintain structural symmetry when combined with other stylistic aspects. In Sec. 3, colored patterns on pre-Columbian textiles from Peru, which have colorings inconsistent with their underlying structural symmetry are discussed.

Journal ArticleDOI
TL;DR: The phenomenon of symmetry in the music of B61a Bart6k has received considerable attention in the critical literature of the past thirty years as mentioned in this paper, and a fascinating variety of information has come to light.
Abstract: The phenomenon of symmetry in the music of B61a Bart6k has received considerable attention in the critical literature of the past thirty years. In the process, a fascinating variety of information has come to light. Symmetry has been depicted as an influence upon structure on many levels, ranging from the smallest to the largest of contexts. The widespread interest in symmetry might well be expected, not only because of its prominence in the audible surface of much of Bart6k's work-a fact that cannot help but provoke speculation about its deeper implications - but also because of the potential power of symmetry to control musical structure. More surprising, perhaps, is that only rarely has symmetry been addressed on its own terms, as an independently functioning system of organization. Rather, theorists have tended to incorporate it into more general frameworks, often designed to assert the primacy of some other dimension of the music.' Of the analytical works that fall into this category, by far the most extensive and probably the most significant is Ern6 Lendvai's B&la Bart6k: An Analysis of His Music, published in English translation in 1971? Many of Lendvai's various models of structure serve to identify symmetrical patterns of durations, chordal functions, and intervals. Symmetry even plays a role in the delineation of form. Lendvai, however, does not attach any special significance to symmetry per se, or to any other single domain of structure,

Journal ArticleDOI
TL;DR: In this paper, four kinds of symmetry models are proposed and their decompositions are given in a square contingency table, two of them are extensions of the local point-symmetry model and the reverse local point symmetry model by TOMIZAWA (1985), and the other two models are concerned with the inclined point symmetry model by Tomizawa (1986) and the conditional symmetric model by MCCULLAGH (1978).
Abstract: In a square contingency table four kinds of symmetry models are proposed and their decompositions are given. Two models of them are extensions of the local point-symmetry model and the reverse local point-symmetry model by TOMIZAWA (1985), and the other two models are concerned with the inclined point-symmetry model by TOMIZAWA (1986) and the conditional symmetry model by MCCULLAGH (1978). An example is given.

Journal ArticleDOI
TL;DR: It is shown that, for symmetry classes of tensors whose associated character has degree higher than one, it is impossible to construct an orthogonal basis of decomposable symmetrized tensors from any basis of the underlying vector space.
Abstract: We present a method for constructing an orthonormal basis for a symmetry class of tensors from an orthonormal basis of the underlying vector space. The basis so obtained is not composed of decomposable symmetrized tensors. Indeed, we show that, for symmetry classes of tensors whose associated character has degree higher than one, it is impossible to construct an orthogonal basis of decomposable symmetrized tensors from any basis of the underlying vector space. We end with an open problem on the possibility of a symmetry class having an orthonormal basis of decomposable symmetrized tensors.

Journal ArticleDOI
TL;DR: Relation “symmetry-system” is investigated from the point of view of the author's version of general systems theory (GST( U )) in the form of the theory of groups of nonevolutionary and evolutionary system transformations and their invariants.
Abstract: Relation “symmetry-system” is investigated from the point of view of the author's version of general systems theory (GST( U )). The symmetry of system is explicated in three ways: (i) in the form of the theory of groups of nonevolutionary and evolutionary system transformations and their invariants, (ii) in the form of a proof of symmetry of the system as itself, (iii) in the form of a proof of the group nature of systems of 2-, 1-, 0-sided actions and relationships. The system of symmetry is described both as a special object-system, and as a specific system of objects of the same kind (in particular, as a system of 64 fundamental and 54 structural symmetries). Premises, the basic concepts of GST( U ), and the laws of system transformations, correspondence, symmetry, and the system similarity are considered in some or other connection. The general theory of isomerism is presented in brief; the relations “isomerism-symmetry”, “system isomorphism-symmetry”, “equality-symmetry” are explicated.

Journal ArticleDOI
01 May 1986-Nature
TL;DR: The concept of symmetry breaking may be expressed mathematically by the reduction of a structure group, for example, choosing a point on a sphere reduces its symmetry group from the group of all rotations to the subgroup of rotations fixing that point.
Abstract: The concept of symmetry breaking may be expressed mathematically by the reduction of a structure group, for example, choosing a point on a sphere reduces its symmetry group from the group of all rotations to the subgroup of rotations fixing that point. A systematic analysis of Penrose's twistor equation1,2 has revealed that twistor theory embodies a truth of nature: that interactive gravitational field theories in space-time correspond to such a symmetry breakdown—a breakdown that occurs, not in space-time, but in twistor space–the space of null geodesies of space-time. In particular, twistor theory lies at the heart of supergravity.

Journal ArticleDOI
TL;DR: In this paper, a program that solves crystal structures completely using Patterson function, symmetry minimum function, atomic minimum superposition and advanced Fourier methods is presented, which works most efficiently for structures with one or several heavier atoms in the asymmetric unit.
Abstract: A program is presented that solves crystal structures completely using Patterson function, symmetry minimum function, atomic minimum superposition and advanced Fourier methods. The program works most efficiently for structures with one or several heavier atoms in the asymmetric unit.

Journal ArticleDOI
TL;DR: Symmetry is analyzed at the atomic level (periodic system, chemical bonding, hybridization), at the molecular level (polyhedral organic molecules and their rearrangement products; isomerism and Polya's theorem; Jahn-Teller effects) and at the supramolecular level (reaction graphs for rearrangements and automerizations; repeating sequences in polymer chains; conservation of orbital symmetry in chemical reactions).
Abstract: Symmetry is analyzed at the atomic level (periodic system, chemical bonding, hybridization), at the molecular level (polyhedral organic molecules and their rearrangement products; isomerism and Polya's theorem; Jahn-Teller effects) and at the supramolecular level (reaction graphs for rearrangements and automerizations; repeating sequences in polymer chains; conservation of orbital symmetry in chemical reactions). A few excursions into mathematics involve solid angles, heuristics of (3, γ)-cages, and visualization of symmetry operations for graphs in three- and four-dimensional Euclidean spaces.



Patent
03 Nov 1986
TL;DR: In this paper, the symmetry of a light reflecting surface was determined by superimposition and comparison of a first reflected image (first image) upon the same image (second image) which has been rotated relative to the first image in a common plane about a common axis.
Abstract: Disclosed is an apparatus and method for determining the symmetry of a light reflecting surface by superimposition and comparison of a first reflected image ("first image") upon the same image ("second image") which has been rotated, relative to the first image, in a common plane about a common axis; wherein such apparatus and method find preferred application in keratometers and keratometry.

Journal ArticleDOI
TL;DR: In this paper, it was shown that the symmetry coefficients and Clebsch-Gordan coefficients can be given in close formulas by using double coset decompositions of the point symmetry group and permutation group.
Abstract: In this paper it is shown that, by use of the double coset decompositions of the point symmetry group and permutation group, the related symmetry coefficients and Clebsch–Gordan coefficients can be given in close formulas.

Journal ArticleDOI
TL;DR: The role and significance of symmetry in the most diverse domains of nature and human activity has been examined by a number of scientists and artists as mentioned in this paper, with a focus on material symmetry.
Abstract: This personal narrative is an introduction to a collective effort by a number of scientists and artists to examine the role and significance of symmetry in the most diverse domains of nature and human activity. Material symmetry, devoid of the rigor of geometrical symmetry, is viewed applicable to material objects as well as abstractions with limitless implications.

Journal ArticleDOI
Dana Wilson1
TL;DR: In this article, the symmetry operations of translation, reflection, and rotation in melodic material are discussed, and the possible reasons for the presence of symmetry in music and the expectation-denial relationship are considered in light of their crucial roles.
Abstract: Symmetry plays a major role in Western music. The first portion of this discussion reveals how the symmetry operations of translation, reflection, and rotation manifest themselves in melodic material. The second portion discusses how symmetry operations are often used by a composer to set up expectation in the listener, only to prevent fulfillment of that expectation through denial of larger-scale symmetrical relations. The possible reasons for the presence of symmetry in music, and of the expectation-denial (“love-hate”) relationship, are considered in light of their crucial roles.

Journal ArticleDOI
TL;DR: In this article, the authors explain the differences between music and its visualization (the score) and through various examples the relation between resonant result and score, and show all the possible concepts of symmetry (and asymmetry) that exist inside this relation and in the graphical signs of certain scores.
Abstract: In this contribution the author attempts to explain the differences between music and its visualization (the score) and through various examples the relation between resonant result and score. In the same time he tries to show all the possible concepts of symmetry (and asymmetry) that exist inside this relation and in the graphical signs of certain scores. A new definition of symmetry is not formulated rather, a direction for a new way to conceive it is given.

Journal ArticleDOI
TL;DR: In this article, the s-classified SU(3) 3j-, 6j- and 9j-symbols are constructed and examined and satisfy simple symmetry relations similar to those valid for the SU(2) case.
Abstract: Starting from the s-classified SU(3) Clebsch-Gordan coefficients introduced previously (Pluhar et al. 1985) the s-classified SU(3) 3j-, 6j- and 9j-symbols are constructed and examined. They satisfy simple symmetry relations similar to those valid for the SU(2) case.