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] (arXiv:1503.05536) 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] tiles in 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 N-gons of the form 8k+j share similar topology and the 8k+2 family satisfies difference equations which generalize the fractal structure of N = 5 & 10. In Appendix II we return to the global web first studied in [SV] (1987) and [H3] (arXiv:1311.6763) and discover that the web W for every regular N-gon can be regarded as the disjoint union of invariant regions defined by pairs of edge-sharing S[k]-S[k'] 'tiles' with k' = N/2-k for N even. At least one index k or k' will be 'primitive' with gcd(k,N) = 1 so the number of such regions is on the order of the algebraic complexity of N.
Categories: math.DS