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

• Published in: Computers & mathematics with applications (1987) 1996, Vol.32 (6), p.101-128
• Main Author: Kohler, A E
• Format: Article
• Language: English
• Subjects: Exact sciences and technology; Form spaces; Geometry; Manifolds; Mathematics; Polygonal forms; Sciences and techniques of general use
• ISSN: 0898-1221; 1873-7668
• DOI: 10.1016/0898-1221(96)00147-2

The geometric structure of isodiastemic manifolds (i.e., manifolds on which the equilaterality of polygons is preserved) within a form space F α spanned by N − 2 consecutive vertex angles α i ( 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 1 The results for N = 5 have been presented in part as poster contributions [1,2].