Q1921225
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. III: Motifs in geometric figures and constraints in form spaces
zbMATH Open document ID: 0932.51007
DOI: 10.1016/0898-1221(96)00046-6
author: Adelhard E. Köhler
publication date: 2000-03-08T00:00:00Z
review text: In these two papers the author continues his former investigations under the same title [Part I: Two and three points, Comput. Math. Appl. 29, No. 3, 63-90 (1995; Zbl 0826.51014)]. The general problem is how to classify geometric figures under a natural group of symmetries. A polygonal form \({\mathcal F}\) specified by a set of \(N\) points in \(E^2\) can be represented by a point in a \(2N\)-dimensional form (shape) space \(\mathbb{F}\). A motif \({\mathcal M}\) within such a form has been defined as a subset \({\mathcal M}\subseteq{\mathcal F}\) on which certain constraints are imposed. The author is influenced, by other natural sciences, e.g. crystallography, he mentions the book [\textit{A. V. Shubnikov} and \textit{V. A. Koptsik}, `Symmetry in science and art', Plenum Press New York (1974)]. Although the reviewer admires author's fanatic efforts, let be repeated here the criticism from the former review: ``\(\dots\) the author applies an ``intuitive terminology'' accepted in the 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. that 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 \(F:=E^2\times E^2\times\cdots\times E^2=:E^{2N}\) and \(G\) is the direct product of the permutation group of \(N\) elements (a finite group of order \(N!)\) and the similarity group (not 0(2)) of the Euclidean plane \(E^2\) (depending on 5 continuous parameters and the sign \(+\) or \(-\) for the determinant of the linear part). The probl…
Mathematics Subject Classification ID: 51M20
Mathematics Subject Classification ID: 57M60
Mathematics Subject Classification ID: 51M05
Mathematics Subject Classification ID: 57N25
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 915128
zbMATH Keywords: classification of shapes
zbMATH Keywords: group actions
reviewed by: Emil Molnár
MaRDI profile type: Publication
full work available at URL: 0898-1221(96)00046-6
OpenAlex ID: W2057662646
published in: Computers & Mathematics with Applications
label: Deformations, isosymmetric manifolds, and higher dimensional form space symmetries for point ensembles (polygonal forms) under \(\mathbb{O}(2)\) symmetry. III: Motifs in geometric figures and constraints in form spaces
description: scientific article; zbMATH DE number 915128