First Families of Regular Polygons and their Mutations

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 an S[k]-S[k’] pair exists it 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]. When the outer-billiards map Tau is introduced in Section 4 the singularity set W will generate two manifolds +W and -W depending on whether N is clockwise or ccw so the relationship between +W and -W depends on how N is related to its mirror image T(N) where T is a horizontal reflection. But N is ‘chiral’ so no rotation by multiples of 2Pi/N can overlay T(N) on N. This means that +W and -W are distinct and W is their ‘union’. But any mapping of +W and -W to W cannot be smooth because their polarities are distinct and we assert that the S[k] dual tiles are the ‘resonances’ of this mapping. Indeed every dual S[k]-S[k’] pair will be split between +W and -W regions so their shared edge defines the boundaries of invariant regions. The Invariance Conjecture of Appendix II says that that every dual S[k] pair with at least k or k’ co-prime with N will survive in W and the number of such pairs is the algebraic complexity C(N) = EulerPhi(N)/2 so every regular N-gon will have C(N) distinct dual pairs and hence C(N)-1 invariant regions. The boundaries of these regions are separatrices which act as eigenstates in a matching Hamiltonian composed of C(N) twist maps.