Deformations, isosymmetric manifolds, and higher dimensional form space symmetries for point ensembles (polygonal forms) under O(2) symmetry II. Four points


AbstractPolygonal forms specified by four points in a plane and their embedding in form spaces F (full eight-dimensional and reduced six-dimensional spaces) are analysed on the lines of Part I of this paper [1]. Isosymmetric manifolds (geometric loci of forms with equal symmetry) in these form spaces are determined. The symmetry of the four-dimensional subspace Fx = (x1, x2, x3, x4) of F is found to be the 48-element group 2405 of four-dimensional crystallography. Its relationship to the symmetry Oh of the three-dimensional subspace F∗x which is orthogonal to the translation Tx is clarified. From the general symmetry operations of form spaces F for N points in En, it is deduced that F must display an n-axial symmetry (for three points in E2, a biaxial symmetry had already been found). Furthermore, the topologies of the isosymmetric manifolds in section planes and in three-dimensional subspaces of the form spaces are discussed. By considering the hierarchy of basis forms and possible transitions between them, conclusions about the manifold topology in E4 can be drawn

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Last time updated on 6/5/2019

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