Showing papers on "Symmetry (geometry) published in 1978"

The Plane Symmetry Groups: Their Recognition and Notation

Abstract: (1978). The Plane Symmetry Groups: Their Recognition and Notation. The American Mathematical Monthly: Vol. 85, No. 6, pp. 439-450.

Orientation and symmetry: effects of multiple, rotational, and near symmetries.

Abstract: Time to detect bilateral symmetry in different orientations was studied for closed polygons with single, double, quadruple, rotational, and near symmetry In Experiment 1, the orientation of the axis of symmetry was varied Detection was fastest for vertical symmetry, next fastest for horizontal, and slowest for left- and right-diagonal symmetries For corresponding orientations, responses were faster to quadruple than double symmetries, and faster to double than single symmetries Negative responses to nearly symmetric figures produced an orientation effect similar to that for single symmetries Rotational symmetries showed no orientation effect and took longer to reject than near symmetries In Experiment 2, subjects looked only for vertical symmetry Responses were twice as fast as in comparable conditions of Experiment 1 The effect of multiple symmetries was still present, but rotational symmetries were rejected faster than near symmetries The results are interpreted as supporting a dual process model for detecting symmetry in multiple orientation channels: Observers first select potential axes of symmetry defined by mirror-similar parts and then evaluate specific axes sequentially in a detailed comparison for mirror-identity

Symmetry and spectroscopy

Abstract: By searching the title, publisher, or authors of guide you truly want, you can discover them rapidly. In the house, workplace, or perhaps in your method can be every best place within net connections. If you direct to download and install the symmetry and spectroscopy, it is agreed simple then, back currently we extend the connect to buy and make bargains to download and install symmetry and spectroscopy for that reason simple!

Ideal vortex motion in two dimensions: Symmetries and conservation laws

Abstract: Each of the conservation laws of the motion of a system of vortices in a two‐dimensional ideal fluid are shown to be uniquely associated with one of the symmetry transformations of the system, one of which can be a scale transformation. In the case of an infinite fluid with no boundaries the five symmetry transformations are: (a) two independent translations in space, (b) spatial rotations, (c) translation in time, and (d) the scale transformation t′ =e2ηt, z′k=eηzk, where η is real and zk are the positions of the vortices in the z plane. Two cases of reduced symmetry are examined: the upper half‐plane and the interior of the unit circle.

Refinement of the margarite structure in subgroup symmetry; correction, further refinement, and comments

Abstract: The crystal structure of margarite-2M1 from Chester County, Pennsylvania, has been refined as an ordered derivative structure in subgroup Cc of the ideal space group C2/ c. Because of the high pseudosymmetry involved, successful refinement by least-squares required initial movement of atomic parameters away from those of the disordered phase towards those of the possible ordered models predicted by a distance least-squares program. Ordering of tetrahedral Si and AI is nearly complete in a pattern that violates centrosymmetry between the two tetrahedral sheets within a 2: I layer. Compositionally similar tetrahedra in adjacent sheets are related instead by a pseudo two-fold axis that extends laterally through the oc­ tahedral AI atoms and is normal to the direction of intralayer shift. The two tetrahedral sheets differ si.in Si,Al contents. A similar ordering pattern is theoretically possible in muscovite-2M, also, but was not found to be adopted by muscovite from the Diamond mine, South Dakota.

Clebsch-Gordan coefficients of symmetry groups

Abstract: A method is given for the determination of Clebsch-Gordan coefficients of finite groups. The Clebsch-Gordan coefficients may be arranged into vectors which are eigenvectors of certain projection matrices. It is shown how an appropriate set of orthonormal eigenvectors of these matrices may be obtained; this set gives then a unitary matrix of Clebsch-Gordan coefficients. The symmetry properties of these Clebsch-Gordan coefficients are studied with special emphasis on the crystallographic space groups.

Isometric group of semi-rigid models: relation to the permutation-inversion group and extension of the concept to non-rigid molecules

Abstract: After a short recapitulation of the basic assumptions underlying the isometric group concept for semi-rigid molecules a relation to the permutation-inversion group (Longuet-Higgins group) is established. Conditions will be put forward under which the permutation-inversion group is homomorphic (not isomorphic) to the isometric group. It will be shown that the familiar symmetry concept of quasi-rigid molecules is identical with the isometric group concept. Finally the latter will be generalized to non-rigid molecules; the isometric group of such systems will be shown to be isomorphic with the isometric group of the associated semi-rigid model. For semi-rigid models with proper covering group the isometric group is shown to be a semi-direct product of the covering group and the internal isometric group.

Erroneous Claims Concerning the Perception of Topological Symmetry

Abstract: A recent issue of this journal carried two dealing directly or indirectly with “inexpensive” (in the sense of low computational effort) methods of perceiving molecular ~ y m m e t r y . ~ More precisely, each article gives a set of rules for scoring2 or comparing) atoms in a molecule with the claim that if the scores are equal, or if the comparison shows no difference, then the atoms are symmetrically equivalent (Le., can be interchanged by some symmetry operation on the molecule). Though both methods are doubtless very good approximations in the sense that they almost always yield the classes of symmetrically equivalent atoms for typical chemical molecules, it is demonstrated below that neither is fully correct. There exist chemical graphs for which the number of distinct symmetry types of atoms is greater than that computed by the suggested algorithms. If these approximate methods are taken as accurate and used, as indicated by Shelley and Munk,2 “. . . in applications of I3C N M R spectroscopy to chemical problems and in the canonical representation and elaboration of molecular structures. . .”, then it must be accepted that in some instances the wrong number of 13C N M R peaks may be predicted, nonunique “canonical” numberings may be created, and correct structures may be overlooked during elaboration. In developing programs of this kind, it is important for one to support each new algorithm for manipulating chemical structures with a solid foundation of graph-theoretic understanding, lest unexpected and perhaps unnoticeable errors occur in the output. Verifying that an algorithm is never known to fail when its output is compared with selected manually generated cases, or with the output of other programs based on totally different principles, does not constitute proof that the algorithm is valid beyond the specific cases which are tested. Such a “proof’ leaves open the important question of the scope and limitations of the algorithm: How far beyond those test cases would one have to look to find one which fails? For this reason, my co-workers and I have, wherever possible, provided rigorous graph-theoretic proofs supporting the methods used in the DENDRAL program^.^,^ It is crucial that readers of this journal understand the approximate nature of the cited symmetry-perception algorithms before applying the methods to their own problems.

R-separation of variables for the four-dimensional flat space laplace and hamilton-jacobi equations

Abstract: All A-separable orthogonal coordinate systems for the complex equations 2/_i3"* » 0 and Sf_i(3,ff)2 = 0 are classified and it is shown that these equations separate in exactly the same systems. Here the complex metric is ds2 = "Z4i_x(dz')2. We show that these two equations separate in precisely the same orthogonal coordinate systems and classify all possibilities. In particular, we show that the /?-separable coor- dinates correspond to coordinates which permit pure separation for the Helmholtz equations on the manifolds EA (flat space), S2 X S2, S3 X Ex and 54 where Sj is the complex ./-dimensional sphere. Detailed group theoretic classifications of separable coordinates on the first three manifolds appear in earlier papers by the authors (1), (2) while an analysis of S4 will appear shortly (3). It follows from these results that each A-separable system {xJ} for (l.l)(a) is characterized by a triplet of second-order commuting symmetry operators {Lx, L2, L3) in the enveloping algebra of 0(6, C), the symmetry algebra of this equation. The .R-separable solutions ¥". of (l.l)(a) corresponding to {xJ} are characterized by the eigenvalue equations Li^ai = ai%j, i =1,2,3,

The structure of higher degree symmetry classes of tensors II

Abstract: Various results bearing on the construction of bases for symmetry classes of tensors are discussed. Applications are given to the production of explicit homogeneous polynomial representations of the full linear group.

Magnetic symmetry, improper symmetry, and Neumann's principle

Abstract: Mathematical tradition has it that transformations characterized by a negative Jacobian determinant are referred to as improper transformations. The symmetry of a physical object corresponding to such an improper transformation becomes an improper symmetry. Improper symmetries have been successfully used for the purpose of crystal symmetry. The extension of these purely spatial symmetries to the domain of spacetime has led to a prejudicial use of light-cone properties, thus affecting adversely an unbiased symmetry classification. We pinpoint these prejudicial procedures and trace their historical origin, while presenting an alternative that restores an unbiased treatment of improper spacetime symmetries. The applications discussed relate to recent developments in the symmetry classification of magnetic crystals and to the extension of Neumann's principle to the time domain.

SU (2s+1) ×ζr and spin statistical weights for tetrahedral XY4, trigonal bipyramidal XY5, and octahedral XY6

Abstract: Using Schur’s theorem we obtain explicit expressions for the spin degeneracies of the three molecular types; XY4 of Td symmetry, XY5 of D3h symmetry, and XY6 of Oh symmetry. Tables of spin statistical weights for all spins up to 11/2 are given.

Cylinders on surfaces

Abstract: (Note that when ~ = 1 the two inequalities coincide.) The proof will consist of two parts: (i) we show the validity in the universal covering of M, /~7/, of the construction given in Figure 3 in [2] (without the symmetry about the vertical geodesic) and then show, as in [2], that the top lateral geodesic in Figure 3 can intersect at most one of the side geodesics; (ii) will then consist of a comparison argument in the universal covering/(/ .

Symmetry adaption: II. Representations on symmetric polyhedra

Abstract: Stimulated by the invention of an algebraic formula for the coefficients of symmetry-adapted linear combinations in a previous paper [1], the representations induced by the edge vectors of a symmetric, molecular polyhedron AmBnCp... are studied in detail. The following objects are defined (in brackets the analogous, common objects): Complete systems of standard functions (spherical harmonics), polyhedral vector coupling coefficients (3jm-symbols), polyhedral Racah coefficients (6j-symbols), polyhedral isoscalar factors (isoscalar factors). The algebraic properties and evaluation methods of these coefficients are discussed in order to be used in subsequent papers on quantities depending on symmetry-adapted orbitals and symmetry coordinates.

Selection rules for the double space group O1h

Abstract: 2014 The octahedral symmorphic space group O1h is the symmetry group of the simple cubic. the caesium chloride, the perovskites and the rhenium trioxide structures. The direct products of irreducible representations of the double space group O1h are reduced for the four points of highest symmetry of the Brillouin zone. LE JOURNAL DE PHYSIQUE TOME 39, MAI 1978, F

The Major-Axis Effect: Axes of Bilateral Symmetry or Loci of Neural Interactions?

Abstract: Distortions in perceptual judgments have been attributed to the axes of bilateral symmetry of surrounding figures In the figures used in prior investigations, the axes of bilateral symmetry have been confounded with loci where heightened neural interaction might be expected to occur In the present experiment, subjects reproduced dots surrounded by figures for which the axes of symmetry and loci of hypothesized neural interactions differed The observed displacements supported the hypothesis that distortions were due to spreading neural effects of the surrounding figures rather than to the effects of axes of bilateral symmetry

Mental rotation of possible and impossible four-cornered toruses

Abstract: Possible and impossible four-cornered toruses were used as stimuli in a mental rotation paradigm. Rotation rates varied with number of “net changes in depth,” a variable that closely corresponds with ratings of impossibility, rather than with formal distinctions between the possible and the impossible. Rotation rates were also affected by perimeter shapes but not in accord with perimeter complexity as defined by number of vertices. Experiment 2 attempted to separate net changes in depth from figure symmetry and found symmetry to be the more reliable predictor of rotation rates, suggesting that symmetry may underlie the influence of net changes of depth.

Effect of rotational symmetry on colored lattices.

Abstract: A colored lattice Lc has a geometrical lattice L. A subgroup lattice L′ of L and each of its cosets consist of like-colored points, each coset having a different color. The index of L′ in L is given by Δ, the determinant of the matrix (tjk) that converts L into L′. This is the order of the factor group {L/L′}, and is also the number n of colors present. The crystal systems—i.e., the combinations of rotational symmetry axes—of Lc, L, and L′ are all the same, but L and L′ may have different centerings; this results in 39 combinations of centerings, called here the 39 symmetry types of colored lattices. These types are tabulated here, together with the special forms taken by (tjk) and the formulas for Δ. In only 7 of the 39 types can the number of colors be arbitrary; in most types certain numbers of colors are impossible.

The Symmetry Potentials of the Ornamental Pages of the Lindisfarne Gospels

Abstract: The information provided in a tenth century colophon leads to speculation that a single hand was involved in the creation of the Lindisfarne Gospels and that, consequently, the book was planned and executed as a unit. The artist may have been Eadfrith, the Irish-trained bishop of Lindisfarne from 698-721. Aldred's colophon mentions only Eadfrith in connection with the text. From this, Bruce-Mitford constructs a persuasive argument identifying scribe and illuminator as one.1 Whether the bishop completed the manuscript before commencing his term or while holding office remains a matter for debate.2 For present purposes it need only be stated that the ornamental pages were composed sometime between Eadfrith's return from Ireland in 690, by which time he had completed his studies and could begin a project of his own, and his death in 721. The five ornamental pages of the Lindisfarne Gospels employ a set of well established decorative motifs traceable through lines of Insular ornament in stone and metal and continental Celtic designs. Yet while drawing on the indigenous Insular sources, the gospels show response to new contact with the Mediterranean world arising from Rome's incursions.3 Each ornamental page is an eloquent blend of \"rational' and \"irrational\" elements developed from the influence of these disparate cultural forces.4 The immediate intention of this paper is to scrutinize the design techniques and the resultant symmetry potentials of the Lindisfarne ornamental pages. Such an approach will serve to emphasize the interplay between Classical and Insular design plans in the late seventh century in respect to design conceptions. An examination of the designs found on the Lindisfarne ornamental pages reveals a self-conscious and deliberate construction and organization of motifs. Although many of the patterns exude the free organic sense characteristic of Celtic art, they were subjected to a rigorously rational scheme. The framework for each pattern was carefully laid out and is today discernible on the recto of the leaf. Here one f1nds visible imprints of regularly spaced prick marks describing the grills and compass designs employed by the painter in building up his decoration. The marks were used to def1ne the area in which the design was to be executed and thus determined the specif1c system of measure to be used for each ornamental detail. As none of these patterns are diapered, it is reasonable to assume that the artist established the recFIGURE 1. Ornamental page from the Lindisfarne Gospels; London, British Library, Cotton MS Nero D. IV, fol. 2V.

[Studies on symmetry in the normal human face using moiré topography (author's transl)].

Abstract: 顔貌の左右対称性は顔面の変形を生ずる各種疾患の診断における重要な指標であるとともに, 顔面の著しい醜形に対する形成手術あるいはエピテーゼによる再建法の発達に伴い治療学上からも検討が要請されてきている。そこで, 立体形状の記録法としてその有用性が広く認められつつあるモアレトポグラフィー法を応用し, 顔面非対称性の計測を試み, まず健常人の生理的範囲内における顔面非対称性の程度, 加齢・性差などによるその変化を検索し, さらに歯牙欠損の与える影響について検討を加えた。研究方法としては健常成人100名および歯牙欠損を有するほかは健常な成人20名を対象とし, 実体格子法によるモアレ写真を作製した。ついでモアレ写真上で, 鼻根・鼻尖・鼻下・上唇・下唇・オトガイ最突出各点の計6個の正中計測基準点を設定した後, 各計測基準点を通る水平断面の左右非対称性を加藤の非対称率として表現した。その結果, 健常人の顔面非対称性ではおおむね非対称率が1桁台であった。非対称率の標準偏差は大きく, 個人差の大きいことをうかがわせた。加齢に伴う変化は, 20歳代と30歳代以上を比較するとすべての計測基準点に関する平均非対称率が後者において前者より大きく, 加齢による顔面非対称性の増大をうかがわせた。性差については, 20歳代男女性の比較で有意差を見いだしがたく, 顔面非対称性は男女性間に差異のないことをうかがわせた。主として上顎前歯・小臼歯部の非対称的歯牙欠損を有する被験者群では, 正常な歯列を有する群より上唇点を通る水平断面の顔面中央部平均非対称率が有意に大であった。