Deformations, isosymmetric manifolds, and higher-dimensional form space symmetries for point ensembles (polygonal forms) under \(\mathbb{O}(2)\) symmetry. I: Two and three points
type: Item
instance of: scholarly article
title: Deformations, isosymmetric manifolds, and higher-dimensional form space symmetries for point ensembles (polygonal forms) under \(\mathbb{O}(2)\) symmetry. I: Two and three points
zbMATH Open document ID: 0826.51014
DOI: 10.1016/0898-1221(94)00229-E
publication date: 1995-03-27T00:00:00Z
review text: The author continues his investigations of the paper in ibid. 25, No. 9, 67-90 (1993; Zbl 0777.92001), which are connected with a general problem, how to classify geometric figures with a general problem, how to classify geometric figures under a natural group of symmetries. E.g. the form space \(F\) for \(N\) points in the Euclidean plane \(E^2\) is a \(2N\)-dimensional Euclidean space under a special similarity group of \(E^{2N}\). By fixing the centre of gravity of the \(N\) points in the origin of the plane \(E^2\), the dimension of the form space reduces to \(2N-2\) to give a reduced form space \(F^*\). Within any form space \(F\), isosymmetric manifolds (geometric loci of all forms having the same symmetry group) and the corresponding symmetry space \(S\) is defined. The author determines in part the symmetry group \(G(S^*)\) of the reduced symmetry space \(S^*\). The four-dimensional symmetry group 21/04, described in the paper, has been found to be a subgroup of the full (non-crystallographic) symmetry group of \(S^*\). The reviewer criticizes that the author applies an ``intuitive terminology'' accepted in chemistry and crystallography but not convenient in a pure mathematical topic where the machinery, e.g. in the theory of Lie transformation groups, has been elaborated for a long time. E.g. the concept of orbit space \(F/G\) is applicable just for the space of essentially different forms, i.e. of shapes. Here \(F\) denotes a convenient parameter space, and \(G\) denotes the group of self-symmetries of \(F\), defined according to the nature of the problem. A point of \(F/G\) is just all the images of a point of \(F\) (the orbit) under the action of the group \(G\). In our case of the \(N\) points in \(E^2\), the parameter space can be chosen as \(F := E^2 \times E^2 \times \dots \times E^2 =: E^{2N}\) and \(G\) is the direct product of the permutation group of \(N\) elements (as a finite group of order \(N!\)) and the similarity group of the Euclidean plan…
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 57N25
zbMATH DE Number: 737506
zbMATH Keywords: classification
zbMATH Keywords: shapes under group actions
zbMATH Keywords: isosymmetric manifolds
zbMATH Keywords: symmetry
author: Adelhard E. Köhler
reviewed by: Emil Molnár
MaRDI profile type: Publication
cites work: Isosymmetric manifolds in form spaces and the normal deformations of polygonal forms
cites work: Q3226733
cites work: Q3232307
cites work: A fuzzy symmetry concept for forms with imperfect symmetries
cites work: Q3259406
cites work: Q3934318
cites work: Q3254275
cites work: Crystallography, geometry and physics in higher dimensions. I. Point-symmetry operations
cites work: Crystallography, geometry and physics in higher dimensions. II. Point symmetry of holohedries of the two hypercubic crystal systems in four-dimensional space
cites work: Crystallography, geometry and physics in higher dimensions. III. Geometrical symbols for the 227 crystallographic point groups in four-dimensional space
cites work: Graphic representation and nomenclature of the four-dimensional crystal classes. II. The individual symmetry operations
cites work: Graphic representation and nomenclature of the four-dimensional crystal classes. III. A notation for the crystal classes
full work available at URL: 0898-1221(94)00229-e
OpenAlex ID: W2076227904
published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 737506