Showing papers on "Symmetry (geometry) published in 1979"
TL;DR: In this article, it was shown that the hidden symmetry of the stationary Einstein equations under the groupSL(2,R) can be extended to a hidden symmetry in the case of Jordan's five-dimensional unified theory.
Abstract: It is shown that the “hidden” symmetry of the stationary Einstein equations under the groupSL(2,R) can be extended to a symmetry under the groupSL(3,R) in the case of Jordan's five-dimensional, unified theory. More generally one obtains an action of the groupSL(n+2,R) on the set of (n+4)-dimensional Einstein spaces admitting a (n+1)-parameter Abelian group of isometries.
122 citations
TL;DR: In this paper, the structure of symmetrical high-angle tilt boundaries with a 〈110〉 axis has been calculated using a potential for aluminium for 3 ⩽ ∑ ⊽ 19.21°.
95 citations
83 citations
TL;DR: In this article, the goodness-of-fit test was proposed and studied over a broad spectrum of alternatives to ellipsoidal symmetry and showed that it has good asymptotic power over a wide range of alternatives.
Abstract: Let $Z$ be a random vector whose distribution is spherically symmetric about the origin. A random vector $X$ which is representable as the image of $Z$ under affine transformation is said to have an ellipsoidally symmetric distribution. The model of ellipsoidal symmetry is a useful generalization of multivariate normality. This paper proposes and studies some goodness-of-fit tests which have good asymptotic power over a broad spectrum of alternatives to ellipsoidal symmetry.
76 citations
TL;DR: In this article, the author explains why the above-mentioned angle is not 90° but 80°, and explains the kind of symmetry involved and the underlying regular tessellations.
Abstract: Of all Escher’s pictures with a mathematical background, the most sophisticated is his 1959 woodcut, Circle Limit III, which uses four colours in addition to black and white. Queues of fishes of each colour are swimming along white arcs that cut the peripheral circle at a certain angle. After discussing the kind of symmetry that is involved and the underlying regular tessellations (so cleverly disguised), the author explains why the above-mentioned angle is not 90° but 80°.
61 citations
TL;DR: In this paper, the class [S] of locally compact groups G is considered, for which the algebra L to the power of 1(G) is symmetric (=Hermitian).
59 citations
TL;DR: Important principles in designing algorithms for topological symmetry perception,vertex-classification, depth-first construction of sequence number permutations, and the use of automorphisms to restrict the construction process are presented.
Abstract: Many nonnumerical computer applications in chemistry require algorithms for topological symmetry perception (constitutional symmetry, the automorphism partitioning problem), the assignment of canonical connection tables (the coding problem), or the detection of graph isomorphism. Important principles in designing such algorithms (vertex-classification, depth-first construction of sequence number permutations, and the use of automorphisms to restrict the construction process) are presented. In addition, a detailed description is presented for the algorithm used in program CASE for the assignment of canonical connection tables and topological symmetry perception.
44 citations
TL;DR: In this article, the symmetry properties of molecules possessing N cyclic degrees of freedom (e.g. torsion angles) are conveniently described in terms of N-dimensional space groups.
Abstract: Symmetry coordinates are useful for describing nuclear arrangements of molecules that can be regarded as being distorted versions of more symmetrical reference structures. The symmetry coordinate description provides a basis for analysing how displacements along particular subsets of symmetry coordinates destroy certain symmetry elements of the reference structure but preserve others (kernel and co-kernel symmetries). It also helps in visualising the symmetry properties of special subspaces of the (3N-- 6)-dimensional internal coordinate space. Some problems concerning the choice of the reference point group are mentioned. It is shown that the symmetry properties of molecules possessing N cyclic degrees of freedom (e.g. torsion angles) are conveniently described in terms of N-dimensional space groups.
44 citations
TL;DR: It is shown that if the vertically and horizontally symmetrical arrays are spatially filtered, so that their respective spectra are 2 octaves apart, then their superposition does not appear random, but both symmetries can be simultaneously perceived.
Abstract: It is known that the sum of a random-dot array with vertical bilateral symmetry and one with horizontal bilateral symmetry appears as a random array. Here we show that if the vertically and horizontally symmetrical arrays are spatially filtered, so that their respective spectra are 2 octaves apart, then their superposition does not appear random, but both symmetries can be simultaneously perceived. The low-band array has a stronger perceptual weight than the high-band array. These demonstrations give further evidence that frequency channels are before symmetry perception.
43 citations
Patent•
23 Nov 1979TL;DR: In this article, a TMS which can convert the output from a pulsed continuum radiation source into a time and wavelength-division multiplexed pulse train is provided, by a single spectrometer.
Abstract: A time-division multiplexed spectrometer (TMS) which can convert the output from a pulsed continuum radiation source into a time- and wavelength-division multiplexed pulse train is provided, by a single spectrometer (1, 2) when: (a) an input source (100) is terminated at the image plane of the spectrometer at a first position which is displaced from the symmetry plane, (b) the first end of a set of optical fibers (100-110), each having a different length, are determined at the image plane at positions which are displaced in the opposite direction from the symmetry plane as is the first position, whereby narrowband portions of the output spectrum are picked up and delayed by different amounts, (c) the second end of the set are terminated at the image plane at positions which are reflections of the terminations of the first end about the symmetry plane, whereby the signals are reinjected into the instrument and refocused onto a second position at the image plane corresponding to the reflection of the first position about the symmetry plane, and (d) an optical receptor is terminated at the second position.
37 citations
TL;DR: In this paper, the authors pointed out that seven crystal structure analyses published in Volume 16 of Inorganic Chemistry (1977) were almost certainly described in space groups of unnecessarily low symmetry, and they carried out successful least-squares refinements in the higher symmetry space groups and have arrived at structures with improved agreement indexes and no unsatisfactory features.
Abstract: It is the purpose of this paper to point out that seven crystal structure analyses published in Volume 16 of Inorganic Chemistry (1977) were almost certainly described in space groups of unnecessarily low symmetry. In two instances the decrease in symmetry resulted in a change of Laue symmetry (from 2/m to 1); since in such cases there is no inherent problem in refinement, the reported molecular geometries are essentially correct, and our reformulations in the higher symmetry result in only minor adjustments. In the remaining five instances, however, the decrease in symmetry involved removing a center of symmetry without changing the Laue group. In such instances, singularities or near-singularities in the least-squares matrices-sometimes noticed by the original authors-led to refinement problems and the resulting structures are in all instances unsatisfactory in some aspects. For four of these latter cases we have carried out successful least-squares refinements in the higher symmetry space groups and have arrived at structures with improved agreement indexes and no unsatisfactory features.
TL;DR: In this article, the authors review and develop geometrical gauging involving the sequence: Lie group/Principal Bundle, for an Internal symmetry group/Soft Group Manifold, for Non-Internal groups.
TL;DR: In this article, the triclinic diffraction symmetry of amesite-2H2 from Antarctica has been shown to be hexagonal with 6-fold biaxial twin sectors on (001), and the twinned crystals produce an average diffraction symmetric that is hexagonal.
Abstract: Cation ordering in amesite-2H2 from Antarctica has reduced the true symmetry from the ideal hexagonal space group P63 to triclinic P1. All crystals show 6-fold biaxial twin sectors on (001), and the twinned crystals produce an average diffraction symmetry that is hexagonal. Individual twin sectors cut from the larger aggregate have 2 V optic angles near 18°, slightly monoclinic unit-cell geometry, and triclinic diffraction symmetry. Structural refinement of an untwinned sector in subgroup symmetry shows nearly complete ordering of Si,Al in tetrahedral sites and of Mg, Al in octahedral sites.
TL;DR: Constraints on two-dimensional half-plane digital transfer functions are developed so that the magnitude responses of these functions possess a specified symmetry (quadrantal symmetry, symmetry about the diagonals, or octagonal symmetry) as discussed by the authors.
Abstract: Constraints on two-dimensional half-plane digital transfer functions are developed so that the magnitude responses of these functions possess a specified symmetry (quadrantal symmetry, symmetry about the diagonals, or octagonal symmetry). The application of these constraints in the design of half-plane filters is indicated.
TL;DR: In this paper, the authors deduce the condition necessary for diffraction enhancement of symmetry to occur in the diffraction pattern of a structure X, and because the symmetry of X coincides with that of its vector set V, the symmetric feature of X derived from the symm of V was studied.
Abstract: An attempt has been made to deduce the condition necessary for diffraction enhancement of symmetry to occur in the diffraction pattern of a structure X, and because the symmetry of the diffraction pattern of X coincides with that of its vector set V, the symmetric feature of X derived from the symmetry of V was studied. The symmetry with the point group G z or Gv/G,,, according as X is inversion-symmetric or not, is defined as the vector symmetry of X, where G V is the point group of V and G, is the inversion group, and when the vector symmetry of X is C n, for example, X is specified as C,-vector-symmetric. When X is homometric with itself by a rotation of 2z#n, it is specified as n-fold self-homometric. X being n-fold self-homometric is the necessary and sufficient condition for X to be n-fold vector-symmetric. Also, X exhibits an enhanced vector (diffraction) symmetry if it is a spacegroupoid structure with the kernel whose point-group symmetry is, other than by addition of an inversion, higher than the point-group symmetry of X. Four examples of enhanced vector symmetry are examined.
TL;DR: The topologic symmetry of a framework silicate is reduced to topochemical symmetry by an ordering inside the tetrahedra; each further reduction of symmetry may be due to one of the following three causes: 1 ordered distribution of extra framework cations, 2 squeezing of the framework, 3 repulsion of extraframework cations as mentioned in this paper.
Abstract: The topologic symmetry of a framework silicate is reduced to topochemical symmetry by an ordering inside the tetrahedra; each further reduction of symmetry may be due to one of the following three causes: 1 ordered distribution of extraframework cations, 2 squeezing of the framework, 3 repulsion of extraframework cations.
Proceedings Article•
20 Aug 1979TL;DR: A system is described that automatically determines the rotational and mirror symmetries of two-dimensional patterns and an implementation based on the SRI vision module is described.
Abstract: A system is described that automatically determines the rotational and mirror symmetries of two-dimensional patterns. The properties of patterns that determine their symmetries are delineated, a representation scheme based on these properties is defined, and algorithms to perform the symmetry analysis are presented. An implementation based on the SRI vision module is described.
TL;DR: In this paper, a complete classification of separable non-orthogonal systems for the flat space Helmholtz equation is given and the relation between separability conditions for the various systems and the classification of abelian sub-algebras of the Euclidean symmetry algebra epsilon (4) is explicitly indicated.
Abstract: A complete classification of separable non-orthogonal systems for the flat space Helmholtz equation is given. The relation between separability conditions for the various systems and the classification of abelian sub-algebras of the Euclidean symmetry algebra epsilon (4) is explicitly indicated.
TL;DR: Some super-figures from [l] have replicating patterns with polar symmetry, and the union of a group of k squares can form the first level of a rep-k construction with four-fold polar symmetry.
TL;DR: The detailed crystal structure of anhydrous aluminum sulfate, Al2(SO4)3, has been determined from X-ray powder diffraction with well crystallized powder sample.
Abstract: The detailed crystal structure of anhydrous aluminum sulfate, Al2(SO4)3, has been determined from X-ray powder diffraction with well crystallized powder sample. The grain was rhombohedron in shape and surrounded by crystal-faces of {0 1 1 2} planes. The unit cell was assigned to hexagonal symmetry with a0=8.032A, c0=21.360A and Z=6, and the space group C3i2-R3.All the atomic parameters were determined by purely geometrical calculations with some assumptions. The calculated intensities of the diffractions from the parameters were in good agreement with the observed ones.Each S atom forms a regular SO4 tetrahedron. The orientation of the SO4 tetrahedron within the unit cell is such that one of the twofold symmetry axes is parallel to the (00l) planes and makes an angle of 16° to one of the a-axis, and the two edges perpendicular to this symmetry axis are inclined at about 80° and 10° respectively to the (00l) planes.Each Al atom is surrounded with the six SO4 tetrahedra and coordinates six O atoms by taking one O atom per one SO4 tetrahedron. The AlO6 octahedra are slightly distorted and classified into two kinds with shape and symmetry.
TL;DR: In this paper, an equation for the roulette in complex polar form was derived for a closed plane curve with points whose x and y coordinates may separately be expanded in Fourier series as functions of the polar angle, assuming these expansions are valid.
Abstract: A large group of closed plane curves may be classified as roulettes, including epicycloids, hypocycloids, and related epitrochoids and hypotrochoids. The equation for the roulette in complex polar form shows that any roulette may be described by the vector sum of two vectors of specified constant magnitudes rotating with constant angular velocities. Methods for plotting, and for electronic display of roulettes are described. An equation is derived for a roulette approximation for an N-sided regular polygon. In particular, an application to two-dimensional potential theory is described and illustrated by consideration of the roulette approximation for a square as an equipotential curve, with derivation of equations for equipotential curves in the field surrounding the square. General equations are derived for given closed plane curves with points whose x and y coordinates may separately be expanded in Fourier series as functions of the polar angle, assuming these expansions are valid. It is shown that, in general, a closed plane curve may be considered as being described by an infinite sum of vectors, each rotating in a circle. Simplifying effects of symmetry about a polar axis and/or about the origin are discussed, and methods for harmonic analysis of a given closed plane curve with aid of an electronic calculator are described.
TL;DR: By utilizing the commutation properties of second quantized operators, a chain of groups has been obtained as discussed by the authors, which can serve the purpose of classifying state of d-electrons in octahedral symmetry.
Abstract: By utilizing the commutation properties of second quantized operators, a chain of groups has been obtained. This chain can serve the purpose of classifying state of d-electrons in octahedral symmetry. The symmetry wave functions, which are adapted to the chain of groups and expressed in the form of linear combinations of Gelfand state, are constructed. Moreover, The methods of calculating matrix elements are discussed systematically in this paper.
Abstract: The objective of this paper is to analyze the behaviour of some minimization methods such as steepest descent method, generalized Newton and quasi-Newton methods under transformations of the variables of the function to be minimized. Energy and molecular coordinates are the function and the variables, respectively, in the case of geometry optimizations. Invariant levels are shown to be decisive for the area where the minimization methods can be successfully employed without rescaling of the coordinates. Specific conditions for symmetry conservation are worked out in context of invariant levels. Symmetry making, breaking and conservation are shown with working examples of geometry optimizations and calculation of energy minimum paths on the basis of certain kinds of molecular coordinates.