Showing papers on "Symmetry (geometry) published in 1990"

The locked rotation function.

Abstract: It frequently occurs that a biological assembly in a crystallographic asymmetric unit has more than one noncrystallographic symmetry operator. For instance, a tetramer might have the point group 222 or a spherical virus will have the point group 532. A self-rotation function searches for the direction and angle of rotation of the individual noncrystallographic symmetry operations, while a cross-rotation function searches for the relationship of a structure in one unit cell with similar structures in another cell. The power of the rotation function can be greatly enhanced by searching for all noncrystallographic symmetry operators simultaneously. The procedure described previously [Rossmann, Ford, Watson & Banaszak (1972). J. Mol. Biol. 64, 237-249] has been generalized. The increased power of this 'locked' rotation function permits a good determination of the orientation of an icosahedral virus in the presence of less than 1% of the possible diffraction data to 7 A resolution. In addition, the peak-to-noise ratio is substantially improved.

Connections: The Geometric Bridge Between Art and Science

Abstract: Proportion in architecture similarity the golden mean graphs tilings with polygons two-dimensional networks and lattices polyhedra - platonic solids transformation of the platonic solids I transformation of the platonic solids II polyhedra - space filling isometries and mirrors symmetry of the plane.

Coordinate frame for symmetry detection and object recognition.

Abstract: Can subjects voluntarily set an internal coordinate frame in such a way as to facilitate the detection of symmetry about an arbitrary axis? If so, is this internal coordinate frame the same as that involved in determining perceived top and bottom in object recognition and shape perception? Subjects were required to determine whether dot patterns were symmetric. Cuing the subjects in advance about the orientation of the axis of symmetry produced a substantial speedup in performance (Experiments 1 and 3) and an increase in accuracy with brief displays (Experiment 2). The effects appeared roughly additive, with an overall advantage for vertical symmetry; thus, the vertical axis effect is not due to a tendency to prepare for the vertical axis. The cuing advantage was found to depend upon the subject's knowing in advance the spatial location as well as orientation of the frame of reference (Experiment 4). The fifth experiment provided evidence that the frame of reference responsible for these effects is the same as the one that determines shape perception: Subjects viewed displays containing a letter (at an unpredictable orientation) and a dot pattern, rapidly naming the letter and then determining whether the dots were symmetric about a prespecified axis. When the top-bottom axis of the letter was oriented the same way as the axis of symmetry for the dots, symmetry judgments were significantly more accurate. Thus, the results suggest a single frame of reference for both types of judgment. The General Discussion proposes a theory of how visual symmetry may be computed, which might account for these phenomena and also characterize their relation to "mental rotation" effects. Language: en

Properties of Binary Relations

Abstract: Summary. The paper contains definitions of some properties of binary relations: reflexivity, irreflexivity, symmetry, asymmetry, antisymmetry, connectedness, strong connectedness, and transitivity. Basic theorems relating the above mentioned notions are given.

Root lattices and quasicrystals

Abstract: It is shown that root lattices and their reciprocals might serve as the right pool for the construction of quasicrystalline structure models. All noncrystallographic symmetries observed so far are covered in minimal embedding with maximal symmetry.

The new ambidextrous universe : symmetry and asymmetry from mirror reflections to superstrings

Abstract: Martin Gardner takes an entertaining look at one of man's most puzzling questions: Is the universe symmetrical? This book is a popular survey of mirror symmetry (left vs. right) and asymmetry, and the significant roles they play in such diverse fields as mathematics, physics, art, music, poetry, and more!

Supermoduli spaces

Abstract: The connection between different supermoduli spaces is studied. It is shown that the coincidence of the moduli space of (1/1) dimensional complex manifolds andN=2 superconformal moduli space is connected with hiddenN=2 superconformal symmetry in the superstring theory.

Shape from Contour Using Symmetries

Abstract: This paper shows that symmetry is essential to shape from contour and indicates problems with existing measures, based on energy and information.

Symbolic solution polynomial equation systems with symmetry

Abstract: Systems of polynomial equations often have symmetry. The Buchberger algorithm which may be used for the solution ignores this symmetry. It is restricted to moderate problems unless factorizing polynomials are found leading to several smaller systems. Therefore two methods are presented which use the symmetry to find factorizing polynomials, decompose the ideal and thus decrease the complexity of the system a lot.In a first approach projections determine factorizing polynomials as input for the solution process, if the group contains reflections with respect to a hyperplane.Two different ways are described for the symmetric group Sm and the dihedral group Dm. While for Sm subsystems are ignored if they have the same zeros modulo G as another subsystem, for the dihedral group Dm polynomials with more than two factors are generated with the help of the theory of linear representations and restrictions are used as well. These decomposition algorithms are independent of the finally used solution technique.We used the REDUCE package Groebner to solve examples from CAPRASSE, DEMARET and NOONBURG which illustrate the efficiency of our REDUCE program. A short introduction to the theory of linear representations is given.In a second approach problems of another class are transformed such that more factors are found during the computation; these transformations are based on the theory of linear representations.Examples illustrate these approaches. The range of solvable problems is enlarged significantly.

Flat points of minimal balance surfaces

Abstract: The symmetry conditions for flat points of minimal surfaces have been studied in relation to the order β of points on such surfaces. Using symmetry aspects, a set of rules for the derivation of fiat points have been developed. By means of these rules the flat points for the 45 families of minimal balance surfaces known so far have been determined. As a check for completeness the relation between the genus of a minimal surface and the orders of its flat points has been used.

3D symmetry-curvature duality theorems

Abstract: We prove theorems showing a duality between the surface curvatures of three-dimensional objects and the existence of symmetry axes. More precisely, we prove that, given a surface, for each maximum or minimum of the principle curvature along a line of curvature, there is a symmetry axis terminating at this point. Moreover, such points are generically the only points at which these axes can terminate. These theorems generalize results obtained by Leyton for two-dimensional objects.

Nonzero star products

Abstract: Many authors have considered the problem of determining necessary and sufficient conditions for the star product of mvectors to be zero. Probably the most remarkable results are those of Merris [9] and Gamas [4]. Our purpose is to generalize these two results to results arbitray symmetry classes of tensors.

Fiber optics bypass switch

Abstract: A bypass switch wherein a spherical reflector is movable between first and second positions relative to an array of optical transceivers, at locations, designated the "S" (source), "D" (detector), "I" (input fiber), "O" (output fiber), and "L1" and "L2" (first and second loop) locations. An optical fiber ("loop fiber") has its ends registered at the L1 and L2 locations. The six optical transceiver terminal locations are characterized by first and second symmetry points. The first symmetry point is midway between the S and O locations and midway between the I and D locations. The second symmetry point is midway the I and O locations, midway between the S and L2 locations, and midway between the L1 and D locations. In the first reflector position, the center of curvature is coincident with the first symmetry point. In the second reflector position, the center of curvature is coincident with the second symmetry point. A simple geometric configuration has the six transceiver locations equidistantly spaced along a line (S, D, I, O, L1, L 2), with the first symmetry point between the D and I locations and the second between the I and O locations.

Symmetry classes of Fokker-Planck-type equations

Abstract: The author has performed a classification of the Fokker-Planck-type equations according to the maximal symmetry groups that keep the equations invariant. It is found that there are only four classes of such equations and the relations between the 'transition probabilities' of the equations have been obtained. He has also obtained complete functional bases of the invariants of the corresponding four groups.

Chiral bismetallocenes with C2 symmetry

Abstract: Synthese de bis-[octohydro biindenyl-1,1' yl] differt a partir de biindenyle-1,1' et de FeCl 2 •2THF. Deux methodes de dedoublement optique ont ete essayees

Quantum symmetry associated with braid group statistics

Abstract: We discuss the concept of “duality” between statistics and symmetry While a compact symmetry group is dual to permutation group statistics, the understanding of the symmetry structure dual to braid group statistics is only at its beginning We use the duality to identify a “first approximation” to this structure and the corresponding algebra of charged fields

Application of coset representations to the construction of symmetry adapted functions

Abstract: A coset representation (G(/Gi)), which is defined algebraically by a coset decomposition of a finite groupG by its subgroupGi, is shown to be a method for the decomposition of a regular body into its point group orbits. This proof also shows that each member of theG(/Gi) orbit belongs to theGi site-symmetry. In addition, a general equation concerning the multiplicities of such coset representations is derived and shown to involve Brester's equations and thek-value equations of framework groups as special cases. The relationship of the coset representation and the site-symmetry affords a general procedure for obtaining symmetry adapted functions.

Symmetry properties in position and momentum space

Abstract: Atomic and molecular symmetry properties for the wave functions and electronic densities in position and momentum spaces are compared. A systematic study of the correspondance of point groups in both spaces is made and examples of experiments of (e, 2e) spectroscopy performed on atoms or molecules are compiled. Simple guidelines for interpreting electron distribution patterns are presented which enable one to sketch the nuclear geometry from the knowledge of momentum maps and Fourier transform effects

Symmetric numerical integral formulas for regular polygons

Abstract: The theory of interpolatory integration formulas in n dimensions is combined with group theory to classify and generate symmetric integration formulas in a systematic way. The points are computed as the common zeros of a set of polynomials. These polynomials constitute a canonical basis for the real ideal belonging to the formula. For a symmetric formula they can be chosen to span a representation of the symmetry group. For regions with high symmetry and formulas of low degree this leads to only a few distinct possibilities and the existence of the corresponding formulas is easily checked. In general, however, the canonical basis is not completely determined by symmetry and degree, and a suitable choice has to be made. The application to regular polygons in two dimensions is discussed and formulas are presented for polygons with three, five, six, seven, and eight vertices, with varying degrees of exactness (up to order 27 for the hexagon).