Richard Evan Schwartz



  • A Textbook Case of Pentagram Rigidity
    preprint (2021) pdf
    companion program

  • The Affine Shape of a Figure Eight Under the Curve Shortening Flow
    preprint (2021) pdf
    (Joint with Matei Coiculescu)

  • The Farthest Point Map on the Regular Dodecahedron
    preprint (2021) pdf
    companion program

  • On The Optimal Embedded Paper Mobius Band
    preprint (2021) pdf
    companion program

  • Paper Moebius bands with T Patterns
    preprint (2020) pdf

  • On Area Growth in Sol
    preprint (2020) pdf

    Research Monographs

  • The Phase Transition in Five Point Energy Minimization
    Research Monograph (2016) pdf
    Companion program download

  • The Plaid Model
    Annals of Mathematics Studies 198 (2018) pdf
    Companion program download

  • The Projective Heat Map
    A.M.S. Research Monograph (2016) pdf
    Companion program download

  • The Octagonal PETs
    A.M.S. Research Monograph (2013) pdf
    Download the companion programs OctaPET and BonePET

  • Outer Billiards on Kites
    Annals of Mathematics Studies 171 (2009)
    the publication
    preprint arXiv version
    companion online material

  • Spherical CR Geometry and Dehn Surgery
    Annals of Mathematics Studies 165 (2007)
    the publication


  • An Improved Bound on the Optimal Embedded Paper Mobius Band
    Geometriae Dedicata (2021) to appear pdf

  • The Spheres of Sol
    (joint with Matei P. Coiculescu)
    Geometry and Topology (2021) to appear pdf
    Companion program download

  • The Farthest Point Map on the Regular Octahedron
    Journal of Experimental Math (2021) to appear pdf
    Companion program download

  • Descartes Circle Theorem, Steiner Porism, and Spherical Designs
    (joint with Serge Tabachnikov) American Math Monthly Vol 127 Issue 3 (2020)

  • A Trichotomy for Rectangles inscribed in a Jordan Loop
    Geometriae Dedicata (2020) to appear pdf

  • Inscribed Rectangle Coincidences
    Advances in Geometry (2019) to appear. pdf

  • Five Point Energy Minimization: A Summary
    Journal of Consructive Approximation (2019) to appear pdf

  • A Hyperbolic View of the Seven Circles Theorem
    Math. Intelligencer (Joint with Kostiantin Drach) to appear (2019) pdf

  • Four Lines and a Rectangle
    J. Exp Math. (2019) to appear pdf
    companion program

  • Square Turning Maps and their Compactifications
    Geometria Dedicata (volume in honor of Bill Goldman's 60th) (2017) pdf

  • Lenghtening a Tetrahedron
    Geometriae Dedicata (to appear) (2014) pdf
    Companion program download

  • The Pentagram Integrals for Poncelet Families
    Journal of Geometry and Physics (2014) pp 384-405 pdf

  • Pentagram Spirals
    J. Exp Math. Vol 22, Issue 4 (2013) pdf
    Companion program click here

  • Liouville-Arnold integrability of the pentagram map on closed polygons
    Duke Mathematics Journal, Vol 162, Number 12 (2012) pp 2149-2196 pdf
    (joint with Valentin Ovsienko and Sergei Tabachnikov )

  • The 5-Electron Case of Thomson's Problem
    J. Experimental Math, vol 22, issue 2, (2013) pp 157-186 pdf
    companion program Electron Champ

  • Outer Billiards on the Penrose Kite: Compactification and Renormalization
    Journal of Modern Dynamics, Issue 3 (2012) pp 473-581 pdf
    Companion applet: The applet
    download the program: the program

  • The Pentagram Integrals for Inscribed Polygons
    Electronic Journal of Combinatorics (2011) to appear
    pdf or companion program (download)
    joint with Sergei Tabachnikov )

  • Outer Billiards and the Pinwheel Map
    Journal of Modern Dynamics Issue 2 (2011) 255-283 pdf
    Companion Applet: Pinwheel

  • The Pentagram map: A discrete integrable system
    Communications in Mathematical Physics 299 (2010) pp 409-446 pdf
    The paper has a small omission. See this erratum or our sequel paper for a fix.
    (joint with Valentin Ovsienko and Sergei Tabachnikov )

  • Elementary Surprises in Projective Geometry
    Mathematical Intelligencer , 2010 (to appear) pdf
    joint with Sergei Tabachnikov )
    companion applet

  • Billiards in Nearly Isosceles Triangles
    Journal of Modern Dynamics, Vol 3 Number 2 (2009) pp 159-231 pdf
    (joint with Pat Hooper)

  • Obtuse Triangular Billiards II: 100 Degrees Worth of Periodic Trajectories
    Journal of Experimental Math , Vol 18, No 2 pp 137-171 (2008)
    paper (.pdf) or wordlist or applet

  • Discrete Monodromy, Pentagrams, and the Method of Condensation
    Journal of Fixed Point Theory and Applications
    (Special edition in Honor of V.I. Arnold's 70th birthday) (2008)

  • Unbounded Orbits for Outer Billiards
    Journal of Modern Dynamics 3 (2007) .pdf
    Research Announcement:
    E.R.A-M.S. Electronic Notices, Vol 14 (2007) web link or .pdf

  • The Poncelet Grid
    Advances in Geometry Vol 7 pp 157-175 (2006)

  • Affine Subdivision, Steerable Semigroups, and Sphere Coverings
    Quarterly Journal of Pure and Applied Mathematics
    (Special volume in honor of G.A. Margulis' 60th Birthday.)

  • Obtuse Triangular Billiards I: Near the (2,3,6) Triangle
    Journal of Experimental Mathematics (2006) Vol 15, Issue 2

  • A Conformal Averaging Process on the Circle
    Geometriae Dedicata Vol 117, Number 1 (2006)
    .pdf or the computer code
    May, 2013: Stergios Antonakoudis found a much faster proof of the
    main inequality in my paper. Click here

  • A Better Proof of the Goldman-Parker Conjecture
    Geometry & Topology Vol 9 (2005) pp 1539-1602
    .pdf or Java applet

  • Modular Circle Quotients and PL Limit Sets
    Geometry&Topology Volume 8 (2004) Paper 1 pp 1-34

  • The Density of Shapes in Three Dimensional Barycentric Subdivision
    Journal of Discrete and Computational Geometry #30 (2003) 373-377

  • Real Hyperbolic on the Outside, Complex Hyperbolic on the Inside
    Inventiones Mathematicae 151.2 (2003) 221-295
    .pdf or computer code

  • Complex Hyperbolic Triangle Groups
    Proceedings of the International Congress of Mathematicians
    (2002) Volume 1: Invited Lectures, pp 339-350

  • Lucy and Lily: a Game of Geometry and Number Theory
    American Math Monthly
    (Jan 2002) Vol 109.1, pp 13-20

  • Circle Quotients and String Art
    Topology (2001) Vol 41.3 pp. 495-523

  • The Pentagram Map is Recurrent
    Journal of Experimental Mathematics
    (2001) Vol 10.4 pp. 519-528

  • Degenerating the Complex Hyperbolic Ideal Triangle Groups
    Acta Mathematica 186
    (2001) Vol 186 p.105-154

  • Ideal Triangle Groups, Dented Tori, and Numerical Analysis
    Annals of Mathematics
    (2001) Vol 153 pp. 533-598
    .pdf or

  • Desargues Theorem, Dynamics and Hyperplane Arrangements
    Geometriae Dedicata
    (2001) Vol 87(1) pp. 261-283

  • Symmetric Patterns of Geodesics and Automorphisms of Surface Groups
    Inventiones Mathematicae
    (1997) Vol 128 pp. 177-199

  • On the Integral Equation of a Third Order O.D.E.
    Journal of Differential Equations
    (1997) Vol 135 pp.183-191

  • Quasi-Isometric Rigidity and Diophantine Approximation
    Acta Mathematica
    (1996) Vol 177 pp. 75-112

  • The Large Scale Geometry of Hilbert Modular Groups
    (with Benson Farb) Journal of Differential Geometry
    (1996) Vol 44 pp. 435-478

  • The Quasi-Isometry Classification of Rank One Lattices
    I.H.E.S. Publications Mathematiques
    (1995) Vol. 82 pp. 133-168

  • The Correlation of Length Spectra on two Hyperbolic Surfaces
    (with Richard Sharp) Communications in Mathematical Physics
    (1993) Vol 153 pp. 323-330

  • Pappus's Theorem and the Modular Group
    I.H.E.S. Publications Mathematiques
    (1993) Vol. 78 pp. 187-206

  • A Projectively Natural Flow for Circle Diffeomorphisms
    Inventiones mathematicae
    (1992) Vol. 110 pp 627-647

  • The Pentagram Map
    Journal of Experimental Mathematics
    (1992) Vol 1 pp. 85-90

  • The Limit Sets of Some Infinitely Generated Schottky Groups
    Transactions of the American Math Society
    (1992) Vol 355 pp. 865-875

    Stuff I Will Probably Never Publish

  • Outer Billiards, Polytope Exchange Transformations, and Quarter Turn Compositions
    preprint (2011) pdf

  • Outer Billiards, The Arithmetic Graph, and the Octagon
    preprint (2010) pdf
    Companion programs: OctoMap 1 and OctoMap 2

  • Outer Billiards, Arithmetic Graphs, and multigrid flows
    (2008) research notes pdf

  • Introduction to McBilliards
    (somewhat outdated) program documentation (2004),
    joint with Pat Hooper
    .pdf or the Mc Billiards program

  • Billiards Obtuse and Irrational,
    light and breezy article (2005) .pdf

  • Dented Tori
    interactive document (1999)
    C/tk program

  • A Dynamical Theorem in Projective Geometry
    interacrive research announcement (1997)

  • Proving Disjointness with Foliated Patches
    speculative technical notes (1999)