First Families of Regular Polygons and their Mutations (1612.09295v12)

Abstract: Every regular N-gon defines a canonical family of regular polygons which are conforming to the bounds of the 'star polygons' determined by N. These star polygons are formed from truncated extended edges of the N-gon and the intersection points determine a scaling which defines the parameters of the family. In H4 we introduced these 'First Families' and here we start from first-principles and devise the geometry of these families independently of any mapping. The first 3 sections focus on this 'star point' geometry and the last 2 sections show that this geometry evolves naturally under the outer-billiards map tau as well as a complex-valued map of A. Goetz. The 'singularity set' W of tau is formed by iterating the extended edges of N and if these edges are truncated when they meet W retains the dihedral symmetry of N. We show that the evolution of W can be reduced to a 'shear and rotation' which preserves the S[k] members of the First Family of N. Lemma 4.1 shows that the center of each S[k] has a constant step-k orbit around N and these 'resonant' orbits establish a connection between geometry and dynamics which is crucial for a study of topology. But except for a few simple cases the topology of W is very complex. We have made some progress in [H6] and [H7] with the 'edge geometry' of N where it appears that N-gons of the form 8k+j share similar topology. In particular we have devised difference equations which describe the edge dynamics for the 8k+2 family and this generalizes the fractal structure of N = 5 & 10. In Appendix II we return to the 'global' web first studied in SV and H3 where the edges of N are not truncated and we show how these extended edges generate endless rings of maximal 'D' tiles where the first ring guarantees that the truncated W is invariant and can serve as a 'template' for the global dynamics.