First Families of Regular Polygons

TL;DR

This paper explores the natural geometric construction of the First Family of regular polygons, linking it to star polygons and cyclotomic fields, with implications for understanding fractal structures and piecewise rotations.

Contribution

It demonstrates how the First Family of regular polygons arises from star polygon geometry and relates to cyclotomic field scales, providing a natural basis for affine piecewise rational rotations.

Findings

First Family scales are derived from star polygon geometry.

The scales form a basis for the maximal real subfield of cyclotomic fields.

Results connect polygon families to fractal and dynamical systems.

Abstract

Every regular polygon can be regarded as a member of a well-defined '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 well-documented for regular N-gons 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 (1290-1349). The nucleus of the First Family are the S[k] 'tiles'. Each S[k] tile has a corresponding 'star-point' 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.