First Families of Regular Polygons
Abstract
Every regular polygon can be regarded as a member of a welldefined 'family' of related regular polygons. These families arise naturally in the study of piecewise rotations such as outer billiards. In some cases they exist on all scales and can be used to define the fractal dimension of the 'singularity set'. This is welldocumented for regular Ngons such as the pentagon, octagon and dodecagon, whose algebraic complexity is 'quadratic' (EulerPhi[N]/2 = 2). Recent evidence suggests that the geometry of these families is intrinsic to the 'parent' polygon and can be derived independently of any mapping. It is the purpose of this paper to show how the First Family for any regular polygon arises naturally from the geometry of the 'star polygons' first studied by Thomas Bradwardine (12901349). The nucleus of the First Family are the S[k] 'tiles'. Each S[k] tile has a corresponding 'starpoint' sk = tan(kPi/N) and a matching scale. Based on a 1949 result by C.Siegel and S.Chowla, the primitive scales with gcd(k,N) = 1 form a unit basis for the maximal real subfield of the cyclotomic field of N. Traditionally, this subfield has been the source of scaling and rotational parameters for affine piecewise rational rotations, and our results show that the First Family scaling can serve as a 'natural' basis for such investigations  which include the outer billiards map.
 Publication:

arXiv eprints
 Pub Date:
 March 2015
 DOI:
 10.48550/arXiv.1503.05536
 arXiv:
 arXiv:1503.05536
 Bibcode:
 2015arXiv150305536H
 Keywords:

 Mathematics  Dynamical Systems;
 37D40;
 37E10;
 52A10;
 52C15;
 97G40;
 00A66;
 51E12;
 53A60;
 37A10
 EPrint:
 New scaling conjecture in Appendix C  with examples. 63 pages