Polygonal figures and their form spaces I. The structure of isodiastemic manifolds (manifolds of equilateral polygons)

The geometric structure of isodiastemic manifolds (i.e., manifolds on which the equilaterality of polygons is preserved) within a form space Fα spanned by N − 2 consecutive vertex angles αi (internal coordinates) of plane N-gons, is reported for N = 5 and N = 6. The curved isodiastemic manifolds are almost everywhere locally (N − 3)-dimensional; exceptions are singular points (linear forms) for even N where the local dimension is N − 2. The isodiastemic manifolds for N = 5 and N = 6 are subdivided into five (N − 3)-dimensional submanifolds of nondegenerate forms (forms without coincident points), each comprising forms with the same angle sum called a form family. The submanifolds are bounded by (N − 4)-dimensional dividing manifolds of degenerate forms which are structured hierarchically according to the types of degeneracy of their forms. For N = 6, the boundary polyhedra for two submanifolds are described in detail: the manifold of spearhead-shaped forms is equivalent to an octahedron, whereas the hexagon manifold is equivalent to a special icosihexahedron.1