  There are multiple mappings that can be used to generate what we call the
'edge geometry' of a regular N-gon, but they are all based on piecewise
isometries acting on the extended edges of N to form a 'singularity' set W.
This singularity set is also known as the 'web' because it is connected and
consists of rays or line segments, with possible accumulation points in the
limit. We will use three such maps here, all of which appear to share the same
local geometry of W. These mappings are the outer-billiards map Tau, the
digital-filter map Df and the 'dual-center' map Dc. In 'Outer-billiards,
digital filters and kicked Hamiltonians' (arXiv:1206.5223) we show that the Df
and Dc maps are equivalent to a 'shear and rotation' in a toral space and the
complex plane respectively, and in 'First Families of Regular Polygons and
their Mutations' (arXiv:1612.09295) we show that the Tau-web W can also be
reduced to a shear and rotation. This equivalence of maps supports the premise
that this web geometry is inherent in the N-gon. Since the topology of W is
complex, we hope to make some progress by studying the region local to N. The
edges of every regular N-gon are part of a Tau-invariant region local to N. The
emphasis here are the S[1] and S[2] First Family tiles adjacent to N, but we
will also study their interaction with neighboring tiles. Since all S[k] tiles
evolve in a multi-step fashion, it is possible to make predictions about the
'next-generation' tiles which survive in the web. The Edge Conjecture defines
just 8 classes of N-gons so there is an 'Eightfold Way' for regular polygons.
