  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.
