arXiv subjects

G. H. Hughes

Publications and source records attributed to G. H. Hughes.

5 recordsLinked to original sources

arXiv2026

First Families of Regular Polygons and their Mutations

Every regular N-gon defines 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 ('star' points) determine the parameters of the family. The First Family Theorem (FFT) shows how each star[k] point defines a matching S[k] 'tile' and the GeneralizedFFT shows how every S[k] can generate their own families with an S[k+1] on the left and (if space permits) an S[k'] on the right where k' = N/2-k for N even.If such a tile does exist S[k] and S[k'] will be called a 'dual' pair and share an edge. When N is even the S[1] tile and N will form a dual pair since N = S[N/2-1] relative to the S[k]. The origin of these dual pairs is the chiral nature of N which implies that the clock-wise and ccw versions of N will generate distinct manifolds +W and -W and the dual pairs will lie on the boundaries with S[k] in one region and S[k'] in the other so their extended shared edge defines the boundary and also a 'primitive' star[k] with gcd(k,N) = 1. For N even, the number of distinct primitive star[k] of N to the right (or left) of the central S[N/2-2] 'M' tile is C(N) = EulerPhi(N)/2 which is also the rank of the maximal real subfield of Q(N). The Twice-Odd Lemma says the M tile of N is a proxy for N/2 so even an odd share manifolds. When the outer-billiards map Tau is introduced in Section 4 this shared edge will form a barrier to the dynamics so it can be extended to define a separatrix Ik which is similar to an 'integral' curve when N is regarded as a harmonic oscillator. So these C(N) Ik curves can be regarded as the resonant eigenstates of N and the matching eigenvalues are the angular 'twists' of the primitive star[k] as defined by the S[k] dual tiles. These same twists define the 'momentum' in a matching Hamiltonian H(N).

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arXiv2025

The Edge Geometry of Regular Polygons -- Part 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 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].

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arXiv2016

First Families of Regular Polygons

Every regular polygon can be regarded as a member of a well-defined 'family' of related regular polygons. These families arise naturally in the study of piecewise rotations such as outer billiards. In some cases they exist on all scales and can be used to define the fractal dimension of the 'singularity set'. This is well-documented for regular N-gons such as the pentagon, octagon and dodecagon, whose algebraic complexity is 'quadratic' (EulerPhi[N]/2 = 2). Recent evidence suggests that the geometry of these families is intrinsic to the 'parent' polygon and can be derived independently of any mapping. It is the purpose of this paper to show how the First Family for any regular polygon arises naturally from the geometry of the 'star polygons' first studied by Thomas Bradwardine (1290-1349). The nucleus of the First Family are the S[k] 'tiles'. Each S[k] tile has a corresponding 'star-point' sk = tan(kPi/N) and a matching scale. Based on a 1949 result by C.Siegel and S.Chowla, the primitive scales with gcd(k,N) = 1 form a unit basis for the maximal real subfield of the cyclotomic field of N. Traditionally, this subfield has been the source of scaling and rotational parameters for affine piecewise rational rotations, and our results show that the First Family scaling can serve as a 'natural' basis for such investigations - which include the outer billiards map.

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arXiv2015

Outer Billiards, Digital Filters and Kicked Hamiltonians

In 1978 Jurgen Moser suggested the outer billiards map (Tangent map) as a discontinuous model of Hamiltonian dynamics. A decade earlier, J.B. Jackson and his colleagues at Bell Labs were trying to understand the source of self-sustaining oscillations in digital filters. Some of the discrete mappings used to describe these filters show a remarkable ability to 'shadow' the Tangent map when the polygon in question is regular. In this paper we describe a specific digital filter map (Df) that appears to have dynamics which are conjugate to the Tangent map for a regular N-gon with N even. When N is odd, there is evidence of another conjugacy between the Tangent map dynamics of N and the matching 2N-gon, so a case like N = 7 can be studied with the Df map in the context of N = 14. This provides a many-fold increase in efficiency, and also allows us to generalize the Tangent map to obtain 'step-k' versions - which have dynamics that are unexplored. We also present some related maps, including Chua and Lin's 3-dimensional version of Df, an Analog to Digital Converter from Orla Feely, a sawtooth version of the Standard Map by Peter Ashwin and various kicked harmonic oscillators. All of these seem to shadow the Tangent map in some form. Mathematica code is provided for all mappings both here and at DynamicsOfPolygons.org.

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arXiv2014

Outer Billiards on Regular Polygons

In 1973, J. Moser proposed that his Twist Theorem could be used to show that orbits of the outer billiards map on a sufficiently smooth closed curve were always bounded. Five years later Moser asked the same question for a convex polygon. In 1987 F. Vivaldi and A. Shaidenko showed that all orbits for a regular polygon must be bounded. R. Schwartz recently showed that a quadrilateral known as a Penrose Kite has unbounded orbits and he proposed that 'most' convex polygons support unbounded orbits. Except for a few special cases, very little is known about the dynamics of the outer billiards map on regular polygons. In this paper we present a unified approach to the analysis of regular polygons - using the canonical 'resonances' which are shared by all regular N-gons. In the case of the regular pentagon and regular octagon these resonances exist on all scales and the fractal structure is well documented, but these are the only non-trivial cases that have been analyzed. We present a partial analysis of the regular heptagon, but the limiting structure is poorly understood and this does not bode well for the remaining regular polygons. The minimal polynomial for the vertices of a regular N-gon has degree Phi(N)/2 where Phi is the Euler totient function, so N = 5, 7 and 11 are respectively quadratic, cubic and quintic. In the words of R. Schwartz, "A case such as N = 11 seems beyond the reach of current technology." Some of the graphics have embedded high-resolution versions so this file is about 39Mb in size. This file and a smaller version can be downloaded at dynamicsofpolygons.org. Just click on PDFs. This paper is dedicated to the memory of Eugene Gutkin (1946-2013) who made fundamental contributions to both inner and outer billiards.

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