arXiv subjects

Richard Evan Schwartz

Publications and source records attributed to Richard Evan Schwartz.

At least 55 records · Page 3Linked to original sources

arXiv2015

The plaid model and outer billiards on kites

This paper is the third in a series which explores a combinatorial method for generating lattice polygons in the plane. I call this method the plaid model. In this paper I prove the main result I had been aiming for since the beginning, which is to show that there is a coarse isomorphism between the plaid model and the so-called arithmetic graph for outer billiards on kites. The content of the theorem is that the plaid model predicts the symbolic dynamics of the outer billiards orbits, up to an error of one unit. This result combines with the work in the other papers to give a second proof that outer billiards has unbounded orbits for every irrational kite. So far, these are the only known polygonal examples with this property.

math.DS↗

arXiv2015

Unbounded orbits for the Plaid Model

This paper is a sequel to my paper "Introducing the Plaid Model". In this paper it is shown that the plaid model has unbounded polygonal paths for every irrational parameter. This result parallels my result that outer billiards on kites has unbounded orbits for every irrational parameter

math.DS↗

arXiv2015

Notes on Shapes of Polyhedra

These are course notes I wrote for my Fall 2013 graduate topics course on geometric structures, taught at ICERM. The notes rework many of proofs in William P. Thurston's beautiful but hard-to-understand paper, "Shapes of Polyhedra". A number of people, both in and out of the class, found these notes very useful and so I decided to put them on the arXiv.

math.GT↗

arXiv2015

Introducing the Plaid Model

We introduce and prove some basic results about a combinatorial model which produces embedded polygons in the plane. The model is closely related to outer billiards on kites, and also is related to corner percolation, to Hooper's Truchet tile system, to self-similar tilings, and to polyhedron exchange transformations.

math.DS↗

arXiv2015

Pan Galactic Division

This is a light expository article. It explains a proof, due to Peter Doyle and Cecil Qiu, the following result from set theory. Let A and B be sets. Suppose there is an injective map from A x {0,...,n-1} into B x {0,...,n-1}. Then there is an injection from A into B. Unlike previous proofs, this one is a very simple one.

math.CO↗

arXiv2014

Unit Lengthenings of Tetrahedra

In this paper we give an affirmative answer to the following question posed by Daryl Cooper: If one lengthens the sides of a tetrahedron by one unit, is the result still a tetrahedron and (if so) does the volume increase? Our proof involves a (presumably) new and sharp inequality involving the Cayley-Menger determinant and one of its directional derivatives. We give a rigorous computer-assisted proof of the inequality. We also sketch an argument which derives the existence portion of the result, in all dimensions, from an old theorem of Von Neumann. Finally, we prove a number of additional results concerning the effect on volume of selectively lengthening some of the sides of a tetrahedron.

math.MG↗

arXiv2013

Linear difference equations, frieze patterns and combinatorial Gale transform

We study the space of linear difference equations with periodic coefficients and (anti)periodic solutions. We show that this space is isomorphic to the space of tame frieze patterns and closely related to the moduli space of configurations of points in the projective space. We define the notion of combinatorial Gale transform which is a duality between periodic difference equations of different orders. We describe periodic rational maps generalizing the classical Gauss map.

math.CO↗

arXiv2013

Pentagram Spirals

We introduce a geometric construction which relates to the pentagram map much in the way that a logarithmic spiral relates to a circle. After introducing the construction, we establish some basic geometric facts about it, and speculate on some of the deeper algebraic structure, such as the complete integrability of the associated dynamical system.

math.DS↗

arXiv2013

Square Turning Maps and their Compactifications

In this paper we introduce some infinite rectangle exchange transformations which are based on the simultaneous turning of the squares within a sequence of square grids. We will show that such noncompact systems have higher dimensional dynamical compactifications. In good cases, these compactifications are polytope exchange transformations based on pairs of Euclidean lattices. In each dimension $8m+4$ there is a $4m+2$ dimensional family of them. Here $m=0,1,2,...$ The case $m=0$, which we studied in depth in an earlier paper, has close connections to the $E_4$ Weyl group and the $(2,4,\infty)$ hyperbolic triangle group.

math.DS↗

arXiv2013

Billiards in Nearly Isosceles Triangles

We prove that any sufficiently small perturbation of an isosceles triangle has a periodic billiard path. Our proof involves the analysis of certain infinite families of Fourier series that arise in connection with triangular billiards, and reveals some self-similarity phenomena in irrational triangular billiards. Our analysis illustrates the surprising fact that billiards on a triangle near a Veech triangle is extremely complicated even though Billiards on a Veech triangle is very well understood.

math.DS↗

arXiv2012

The Octagonal PET I: Renormalization and Hyperbolic Symmetry

We introduce a family of polytope exchange transformations (PETs) acting on parallelotopes in $\R^{2n}$ for $n=1,2,3...$. These PETs are constructed using a pair of lattices in $\R^{2n}$. The moduli space of these PETs is $GL_n(\R)$. We study the case n=1 in detail. In this case, we show that the 2-dimensional family is completely renormalizable and that the $(2,4,\infty)$ hyperbolic reflection triangle group acts (by linear fractional transformations) as the renormalization group on the moduli space. These results have a number of geometric corollaries for the system. Most of the paper is traditional mathematics, but some part of the paper relies on a rigorous computer-assisted proof involving integer calculations.

math.DS↗

arXiv2012

The Octagonal PET II: The Topology of the Limit Sets

This is a sequel to my paper "The Octagonal PET I: Renormalization and Hyperbolic Symmetry". In this paper we use the renormalization scheme found in the first paper to classify the limit sets of the systems according to their topology. The main result is that the limit set is either a finite forest or a Cantor set, with an explicit description of which cases occur for which parameters. In one special case, the limit set is a disjoint union of 2 arcs if and only if the continued fraction expansion of the parameter has the form [a0:a1:a2:a3...] with a_k even for every odd k.

math.DS↗

arXiv2011

Liouville-Arnold integrability of the pentagram map on closed polygons

The pentagram map is a discrete dynamical system defined on the moduli space of polygons in the projective plane. This map has recently attracted a considerable interest, mostly because its connection to a number of different domains, such as: classical projective geometry, algebraic combinatorics, moduli spaces, cluster algebras and integrable systems. Integrability of the pentagram map was conjectured by R. Schwartz and later proved by V. Ovsienko, R. Schwartz and S. Tabachnikov for a larger space of twisted polygons. In this paper, we prove the initial conjecture that the pentagram map is completely integrable on the moduli space of closed polygons. In the case of convex polygons in the real projective plane, this result implies the existence of a toric foliation on the moduli space. The leaves of the foliation carry affine structure and the dynamics of the pentagram map is quasi-periodic. Our proof is based on an invariant Poisson structure on the space of twisted polygons. We prove that the Hamiltonian vector fields corresponding to the monodoromy invariants preserve the space of closed polygons and define an invariant affine structure on the level surfaces of the monodromy invariants.

math.DS↗

arXiv2011

Outer Billiards on the Penrose Kite: Compactification and Renormalizaiton

In this long paper we give a fairly complete analysis of outer billiards on the Penrose kite. Our analysis reveals that this 2-dimensional non-compact system has a 3-dimensional compactification, a certain polyhedron exchange map, and that this compactification has a renormalization scheme. These two features allow us to make some sharp statements concerning the distribution, large-scale geometry, fine-scale geometry, and hidden algebraic symmetries of the orbits. For instance, one of our results is that the union of the unbounded orbits has Hausdorff dimension 1. We give a computer-aided proof of the results concerning the compactification and the renormalization. This proof involves finitely many calculations done with exact integer arithmetic.

math.DS↗

arXiv2010

Outer Billiards, Arithmetic Graphs, and the Octagon

Outer Billiards is a geometrically inspired dynamical system based on a convex shape in the plane. When the shape is a polygon, the system has a combinatorial flavor. In the polygonal case, there is a natural acceleration of the map, a first return map to a certain strip in the plane. The arithmetic graph is a geometric encoding of the symbolic dynamics of this first return map. In the case of the regular octagon, the case we study, the arithmetic graphs associated to periodic orbits are polygonal paths in R^8. We are interested in the asymptotic shapes of these polygonal paths, as the period tends to infinity. We show that the rescaled limit of essentially any sequence of these graphs converges to a fractal curve that simultaneously projects one way onto a variant of the Koch snowflake and another way onto a variant of the Sierpinski carpet. In a sense, this gives a complete description of the asymptotic behavior of the symbolic dynamics of the first return map. What makes all our proofs work is an efficient (and basically well known) renormalization scheme for the dynamics.

math.DS↗

arXiv2010

The Pentagram Integrals on Inscribed Polygons

The pentagram map is a natural iteration on projective equivalence classes of (twisted) n-gons in the projective plane. It was recently proved ([OST]) that the pentagram map is completely integrable, with the complete set of Poisson commuting integrals given by the polynomials O1,...,O[n/2],On and E1,...,E[n/2],En, previously constructed in [S3]. These polynomials are somewhat reminiscent of the symmetric polynomials. It was observed in computer experiments that if a polygon is inscribed into a conic then Oi=Ei for all i. The goal of the paper is to prove this theorem. The proof is combinatorial, and it was also suggested by computer experimentation.

math.CO↗

arXiv2010

Outer Billiards and the Pinwheel Map

In this paper we establish a kind of bijection between the orbits of a polygonal outer billiards system and the orbits of a related (and simpler to analyze) system called the pinwheel map. One consequence of the result is that the outer billiards system has unbounded orbits if and only if the pinwheel map has unbounded orbits. As the pinwheel map is much easier to analyze directly, we think that this bijection will be helpful in attacking some of the main questions about polyonal outer billiards.

math.DS↗

arXiv2010

The 5 Electron Case of Thomson's Problem

We give a rigorous computer-assisted proof that the triangular bi-pyramid is the unique configuration of 5 points on the 2-sphere that globally minimizes the Coulomb (1/r) potential. We also prove the same result for the (1/r^2) potential. The main mathematical contribution of the paper is a fairly efficient energy estimate that works for any number of points and any power-law potential.

math.MG↗