Edge Geometry of Regular Polygons -- Part 2

TL;DR

This paper explores the complex edge geometry of regular polygons through mappings like outer-billiards, digital filters, and dual-center maps, revealing inherent web structures and local invariants that suggest a universal geometric framework.

Contribution

It demonstrates the equivalence of different edge mappings to shear and rotation transformations, and studies local invariant regions to understand the web topology of regular polygons.

Findings

Mappings are equivalent to shear and rotation transformations.

The web W has a complex topology with local invariant regions.

Predictions about next-generation tiles in the web are possible.

Abstract

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.