First Families of Regular Polygons and their Mutations

TL;DR

This paper explores the geometric and dynamical properties of regular polygons, their star polygon families, and their invariance under outer-billiards maps, revealing deep links between geometry and dynamics.

Contribution

It introduces a new framework connecting star polygons, tilings, and invariant regions under the outer-billiards map Tau, proposing a potential KAM-like structure for regular polygons.

Findings

Star[k] points define scale and tiles independent of mappings.

S[k] tiles are preserved by the outer-billiards map Tau.

Conjecture of multiple invariant regions and possible KAM structures.

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 ('star' points) determine a scaling which defines the parameters of the family. In the first 3 sections we use these star[k] points to define the scale[k] and matching S[k] 'tiles' independent of any mapping. In Section 4 we introduce the outer-billiards map Tau and show that the S[k] First Family tiles are preserved by the singularity set W, so these are fundamental 'resonances' of Tau. At the next level every star[k] defines a (Schlafli) {N,k} star-polygon orbit that skips k vertices of N so it has period N/gcd(k,N) and Lemma 4.1 defines a duality between the star[k] and cS[k] so the S[k] have the same period. This is a critical link between geometry and dynamics.Since the star[k] are in W these {N,k} star polygons are also in W and they define a framework for invariant regions. In Appendix II we show that each Tau-invariant region is bounded by pairs of 'dual' S[k]-S[k'] tiles where k' = N/2-k for N even. The Edge Sharing Lemma shows that these tiles share an edge and this shared edge is aligned with a star point of N. A 1949 result of C.L.Siegel states that the primitive star[k] with gcd(k,N) = 1 are independent and they have the same rank as the vertices of N which is C(N) = EulerPhi[N]/2. The Invariance Conjecture states that all regular N-gons will have C(N)-1 distinct invariant regions in W. There is a surprising degree of correlation between these invariant regions and the traditional KAM 'islands' so there may be a KAM Theorem for regular polygons where modified {N,k} play the role of separatrices and a form of KAM 'cantori' might survive in the 'residual set' of W which is conjectured to have zero-measure.