3 years ago

First Families of Regular Polygons and their Mutations.

G.H.Hughes, arXiv:1503.05536

Every regular N-gon generates 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 determine a scaling which defines the parameters of the family. In 'First Families of Regular polygons' (arXiv:1503.05536) we showed that this scaling forms a basis for S(N), the maximal real subfield of the cyclotomic field of N. The traditional generator for this subfield is 2cos(2Pi/N) so it has order Phi(N)/2 where Phi is the Euler totient function. This order is known as the 'algebraic complexity' of N. The family of regular polygons shares this same scaling and complexity, so members of this family are an intrinsic part of any regular polygon - and we call them the First Family of N.

Because the First Family members are also regular polygons, their families can be used to define a recursive geometry with known scaling. This would typically lead to a multi-fractal topology for the star polygons of N, and the scaling parameters would be algebraic units in S(N) so calculations would be efficient and exact. This scenario actually exists under a piecewise isometry such as the outer-billiards map T, and here we investigate the connection between the First Families of N and the 'singularity' set formed by iterating the extended edges of N under T.

Publisher URL: http://arxiv.org/abs/1612.09295

DOI: arXiv:1612.09295v5

You might also like
Discover & Discuss Important Research

Keeping up-to-date with research can feel impossible, with papers being published faster than you'll ever be able to read them. That's where Researcher comes in: we're simplifying discovery and making important discussions happen. With over 19,000 sources, including peer-reviewed journals, preprints, blogs, universities, podcasts and Live events across 10 research areas, you'll never miss what's important to you. It's like social media, but better. Oh, and we should mention - it's free.

  • Download from Google Play
  • Download from App Store
  • Download from AppInChina

Researcher displays publicly available abstracts and doesn’t host any full article content. If the content is open access, we will direct clicks from the abstracts to the publisher website and display the PDF copy on our platform. Clicks to view the full text will be directed to the publisher website, where only users with subscriptions or access through their institution are able to view the full article.