Deformations, isosymmetric manifolds, and higher dimensional form space symmetries for point ensembles (polygonal forms) under (2) symmetry II. Four points

Polygonal forms specified by four points in a plane and their embedding in form spaces (full eight-dimensional and reduced six-dimensional spaces) are analysed on the lines of Part I of this paper [1]. Isosymmetric manifolds (geometric loci of forms with equal symmetry) in these form spaces are determined. The symmetry of the four-dimensional subspace of is found to be the 48-element group 24/05 of four-dimensional crystallography. Its relationship to the symmetry h of the three-dimensional subspace *x which is orthogonal to the translation Tx is clarified. From the general symmetry operations of form spaces for N points in n, it is deduced that must display an n-axial symmetry (for three points in 2, a biaxial symmetry had already been found). Furthermore, the topologies of the isosymmetric manifolds in section planes and in three-dimensional subspaces of the form spaces are discussed. By considering the hierarchy of basis forms and possible transitions between them, conclusions about the manifold topology in 4 can be drawn.