A.E. Köhler
University of Jena
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Computers & Mathematics With Applications | 1993
A.E. Köhler
Abstract Form variations are described in an appropriately constructed form space F (typically an R n ), where every point of F represents a different form. Regarding the symmetries of the forms, F can be divided into disjunct isosymmetric manifolds, i.e., points, lines, surfaces, and volumes whose points correspond to forms with equal symmetries. These manifolds are derived from a symmetry analysis of possible deformations of the forms. This analysis is comparable to the construction of symmetry coordinates in a normal coordinate analysis of molecules and results in normal modes of deformation (“normal deformations”) of these forms. From the symmetry species of a normal deformation, the symmetry of the resulting form can be inferred. Transformation of the form space coordinates into normal coordinates (the differentials of which are the normal deformations) facilitates the description of the high-dimensional form spaces and can be made the basis of an easy symmetry diagnosis of forms. Furthermore, the problem of an ascent in symmetry by deformation is discussed.
Computers & Mathematics With Applications | 1993
A.E. Köhler
Abstract A second approach to the evaluation of membership functions for a fuzzy form symmetry concept (Kohler, 1991) is described. Fuzzification of the symmetry requirements (only approximate identity or even complete non-identity instead of identity original and transformed object) and measurement of the degree of identity between original and transformed object opens a novel way for the evaluation of the degree of symmetry of an object. Formulae are given for both discrete and continuous form functions R ( θ ). Furthermore, symmetry profiles of forms for different symmetry operations (rotation, reflection, inversion and glide-reflection) are developed, which allow a subtle characterisation of the symmetry properties of forms and their variation during form transitions. The symmetries of the symmetry profiles are correlated with the symmetries of the corresponding form functions, for which a classification is proposed.
Computers & Mathematics With Applications | 1996
A.E. Köhler
A polygonal form F specified by a set of N points in En (here: a plane) can be represented by a point in an n · N-dimensional form space F. A motif M within such a form has been defined as a subset M ⊆ F on which certain constraints are imposed. Different types of constraints are discussed. In any case, every independent constraint reduces the dimension of the form space by one so that the resulting forms correspond to points in a constrained form space Fc ⊂ F. The subspaces Fc are investigated for some forms containing rigid and—as a novel aspect—deformable motifs, and they are correlated with the full form spaces F described in previous papers [1–3]. Furthermore, constraints which do not lead to motifs are treated. Generally, constraints of distances define circles in certain planes of F so that angular coordinates have to be used. As a special case, cyclic polygons have been found to lie on hyperspheres in F. On the other hand, if the distances between points are constrained to such an extent that only two degrees of freedom (the global rotation and one degree of shape variation) are left, then the resulting forms have been found to lie on hyperellipsoids in F. These constitute special cases of isodiastemic manifolds consisting of forms with fixed point distances.
Computers & Mathematics With Applications | 1995
A.E. Köhler
The analysis of polygonal forms and their form transitions using normal deformations [1] has been extended to a global analysis of form spaces for polygonal forms specified by N points in a plane. Coincidences of the points are explicitly allowed, and the origin of the form space is taken to correspond to a coincidence of all N points. This choice is natural since for every form there exists a totally symmetric deformation leading to exactly this configuration. The form space F for N points in a plane is a 2N-dimensional Euclidean space whose points represent all possible N-vertex polygons (simple and self-intersecting ones). By fixing the centres of gravity of the forms in the origin of the x, y plane, the dimension of the form space reduces to 2N − 2 to give a reduced form space F∗. Within the form spaces, isosymmetric manifolds (geometric loci of all forms having the same symmetry G) are determined. These manifolds inscribed in a form space F define the corresponding symmetry space S. The symmetry G (S∗) of the reduced symmetry space for three points has been determined in part. The four-dimensional symmetry group 21/04 has been found to be a subgroup of the full (noncrystallographic) symmetry of S∗. For some cases, the relation G (F∗) ⊆ G (S∗) has been exemplified.
Computers & Mathematics With Applications | 1994
A.E. Köhler
Abstract The notions of motif and arrangement symmetries within composite geometric figures are defined. The relationships between these types of symmetry and the symmetry of the whole figure are clarified by making use of the crystallographic concepts of site symmetry and direction symmetry. From this, it has been deduced that a figure with arbitrary symmetry can be assembled from motifs of likewise arbitrary symmetries. If a motif with symmetry G M is placed on a site having the site symmetry G S ⊆ G M , its contribution to the figure symmetry G is only a subgroup G * MO of its direction symmetry G MO where G S = G * MO ⊆ G MO ⊆ G M . Supernumerary symmetry elements of the motif give rise to intermediate or latent symmetries of the figure. A consequent decomposition of a geometric figure into its constituent points reveals that a large part of the O ( n ) symmetry of every single point is lost when assembling these points to build up the figure. All “lost” symmetries can, however, be detected as intermediate symmetries of this figure. They can be displayed as fuzzy symmetry landscapes and symmetry profiles for a given figure showing all crisp and intermediate symmetries of interest.
Computers & Mathematics With Applications | 1996
A.E. Köhler
Abstract The geometric structure of isodiastemic manifolds (i.e., manifolds on which the equilaterality of polygons is preserved) within a form space F α spanned by N − 2 consecutive vertex angles α i ( internal coordinates ) of plane N -gons, is reported for N = 5 and N = 6. The curved isodiastemic manifolds are almost everywhere locally ( N − 3)-dimensional; exceptions are singular points (linear forms) for even N where the local dimension is N − 2. The isodiastemic manifolds for N = 5 and N = 6 are subdivided into five ( N − 3)-dimensional submanifolds of nondegenerate forms (forms without coincident points), each comprising forms with the same angle sum called a form family . The submanifolds are bounded by ( N − 4)-dimensional dividing manifolds of degenerate forms which are structured hierarchically according to the types of degeneracy of their forms. For N = 6, the boundary polyhedra for two submanifolds are described in detail: the manifold of spearhead-shaped forms is equivalent to an octahedron, whereas the hexagon manifold is equivalent to a special icosihexahedron. 1
Computers & Mathematics With Applications | 2000
A.E. Köhler
Abstract In a closed nonregular equilateral polygonal curve, angles cannot be permuted arbitrarily without opening the curve. Here we investigate the possible angle permutations that leave a polygonal curve closed. To this end we introduce angle sequences , i.e., symbol sequences describing the number and arrangement of like angle values in classes of forms. We find that there exist “ universal angle sequences ” giving closed polygons for arbitrary angle values provided that the angle sum is kept constant at one of certain permissible values. The universal angle sequences are periodic : they consist of at least two identical series of symbols. Within a series, the symbols can be permuted freely on condition that in all other series the very same permutation is applied. All these correlated permutations , when applied to the angles of a corresponding polygonal curve, leave this curve closed. Polygons belonging to a given angle sequence (form class) constitute a submanifold of the manifold M I of equilateral polygons within the form space. These submanifolds structure M I in a hierarchical manner so that the form class concept gives a valuable complementary classification scheme for the forms of M I . Furthermore, we find that polygons with constant angle composition (the term “composition” is used here in its chemical sense) show a wide variety of possible shapes (from quasi-circular to wormlike and even self-intersecting ones) corresponding to different degrees of “segregation” of like angles within the polygonal curve.
Computers & Mathematics With Applications | 1997
A.E. Köhler
Abstract The fuzzy symmetry concept allows, for plane geometric figures, a separate characterisation of the degrees of C n symmetry (polymetry) and C nv symmetry (polysymmetry). A new method for the evaluation of these degrees of symmetry δ from the areas of the maximal C n and C nv regions within a figure is presented. Since δ depends on the rotation point position, symmetry profiles and symmetry landscapes of a figure can be calculated which give a detailed local characterisation of the figure with respect to arbitrary symmetries. From such landscapes, symmetry centroids of a figure are determined. It is shown that the different symmetry centroids do not coincide in figures with low symmetry (C 1 , C 1 v ).
Zeitschrift für Chemie | 2010
Peter Fink; Walter Pohle; A.E. Köhler
Zeitschrift für Chemie | 2010
Harald Winde; Peter Fink; A.E. Köhler