First Families of Regular Polygons and their Mutations

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 the parameters of the family. The First Family Theorem (FFT) shows how each star[k] point defines a matching S[k] 'tile' and the GeneralizedFFT shows how every S[k] can generate their own families with an S[k+1] on the left and (if space permits) an S[k'] on the right where k' = N/2-k for N even.If such a tile does exist S[k] and S[k'] will be called a 'dual' pair and share an edge. When N is even the S[1] tile and N will form a dual pair since N = S[N/2-1] relative to the S[k]. The origin of these dual pairs is the chiral nature of N which implies that the clock-wise and ccw versions of N will generate distinct manifolds +W and -W and the dual pairs will lie on the boundaries with S[k] in one region and S[k'] in the other so their extended shared edge defines the boundary and also a 'primitive' star[k] with gcd(k,N) = 1. For N even, the number of distinct primitive star[k] of N to the right (or left) of the central S[N/2-2] 'M' tile is C(N) = EulerPhi(N)/2 which is also the rank of the maximal real subfield of Q(N). The Twice-Odd Lemma says the M tile of N is a proxy for N/2 so even an odd share manifolds. When the outer-billiards map Tau is introduced in Section 4 this shared edge will form a barrier to the dynamics so it can be extended to define a separatrix Ik which is similar to an 'integral' curve when N is regarded as a harmonic oscillator. So these C(N) Ik curves can be regarded as the resonant eigenstates of N and the matching eigenvalues are the angular 'twists' of the primitive star[k] as defined by the S[k] dual tiles. These same twists define the 'momentum' in a matching Hamiltonian H(N).