The Edge Geometry of Regular Polygons -- Part 1

TL;DR

This paper explores the complex edge geometry of regular polygons through mappings like outer-billiards and digital filters, revealing an inherent web structure and classifying polygons into an 8-fold categorization.

Contribution

It introduces algebraic criteria for invariance of edge regions and extends conjectures on local web structures, connecting various mappings and polygon classifications.

Findings

Identifies 8 classes of regular N-gons forming an '8-Fold Way'

Establishes equivalence of different edge-mappings to shear and rotation

Provides algebraic criteria for invariance of edge-sharing regions

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 of Chou and Lin and a 'dual-center' map of Arek Goetz. In (arXiv:1206.5223) we show that these maps are equivalent to a 'shear and rotation' in a toral space and the complex plane respectively, and in the main paper [H5] '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. In terms of Edge Geometry there appears to be just 8 classes of regular N-gons which make up an '8-Fold Way'. Since the topology of W is very 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 that should include at least 1/4 of the First Family S[k] tiles of N, so the edge geometry overlaps the global web. In Appendix II to [H5] we formulated what may be sufficient conditions for invariance of these regions local to N, based on 'edge-sharing' of adjacent tiles. In this paper we extend this conjecture and provide algebraic criteria with many examples. In the appendix we give examples of 'projections' of orbits based on the work of R.Schwartz in [S2].