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
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 problem seems to be very difficult, in general, but for small \(N\) the computer always helps.'' The author should formulate his problems to computer specialists otherwise his enthusiastic work seems to be hopeless. The author develops remarkable ideas for simplifications of such a program. [For part II: `Four points', ibid. 31, No. 9, 89-116 (1996) see the paper above].
scientific article; zbMATH DE number 915128