Deformations, isosymmetric manifolds, and higher-dimensional form space symmetries for point ensembles (polygonal forms) under symmetry I. Two and three points

The analysis of polygonal forms and their form transitions using normal deformations [1] has been extended to a global analysis of form spaces for polygonal forms specified by N points in a plane. Coincidences of the points are explicitly allowed, and the origin of the form space is taken to correspond to a coincidence of all N points. This choice is natural since for every form there exists a totally symmetric deformation leading to exactly this configuration. The form space for N points in a plane is a 2N-dimensional Euclidean space whose points represent all possible N-vertex polygons (simple and self-intersecting ones). By fixing the centres of gravity of the forms in the origin of the x, y plane, the dimension of the form space reduces to 2N − 2 to give a reduced form space . Within the form spaces, isosymmetric manifolds (geometric loci of all forms having the same symmetry ) are determined. These manifolds inscribed in a form space define the corresponding symmetry space . The symmetry of the reduced symmetry space for three points has been determined in part. The four-dimensional symmetry group 21/04 has been found to be a subgroup of the full (noncrystallographic) symmetry of . For some cases, the relation has been exemplified.