The Edge Geometry of Regular Polygons -- Part 1

G. H. Hughes

Published 2021-03-11Version 1

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 in the complex plane respectively, and in 'First Families of Regular Polygons and their Mutations' (arXiv:1612.09295) we show that the web for Tau 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. Here we describe the edge geometry up to N = 25 and in Part 2 this will be extended to N = 50. In all cases this geometry defines an invariant region local to N. Typically this region contains multiple S[k] 'tiles' from the First Family of N, but our emphasis is on the S[1] and S[2] tiles adjacent to N. Since the web evolves in a multi-step fashion, it is possible to make predictions about the 'next-generation' tiles which will survive in the early web of S[1] and S[2]. The Edge Conjecture defines just 8 classes of N-gons based on this edge geometry. Since the webs are recursive these predictions have long-term implications.