Deformations, isosymmetric manifolds, and higher-dimensional form space symmetries for point ensembles (polygonal forms) under \(\mathbb{O}(2)\) symmetry. I: Two and three points
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 plane \(E^2\) (depending on 5 continuous parameters and a discrete one: + or \(-\) for the sign of the determinant of the linear part). The problem seems to be very difficult, in general, but for small \(N\) the computer always helps.
scientific article; zbMATH DE number 737506
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