Richard Evan Schwartz



  • Continued Fractions and the 4-Color Theorem
    preprint (2022) pdf

  • Pentagram Rigidity for Centrally Symmetric Octagons
    preprint (2021) pdf
    companion program

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

  • 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

  • Polygonal Symplectic Billiards
    (joint with Peter Albers, Guatam Banhatti, Filip Sadlo, Serge Tabachnikov)
    preprint (2019) arXiv version

    Research Monographs

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

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

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

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

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

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


    1. The affine shape of a figure 8 curve under the curve shortening flow
      (joint with Matei P. Coiculescu)
      Journal of Differential Geometry (2022) to appear pdf

    2. Trisecting the 9-vertex complex projective plane
      Math. Intelligencer (2022) to appear pdf

    3. On Projective Evolutes of Polygons
      Experimental Mathematics (2022) to appear pdf
      companion program
      (Joint with Maxim Arnold and Serge Tabachnikov)

    4. Conway's Nightmare: Brahmagupta and Butterflies
      Math. Intelligencer (2022) to appear pdf

    5. Survey Lecture on Billiards
      Proceedings of the 2022 ICM (2022) to appear pdf

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

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

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

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

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

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

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

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

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

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

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

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

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

    19. 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 )

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

    21. 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

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

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

    24. 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 )

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

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

    27. 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

    28. 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)

    29. 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

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

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

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

    33. 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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

    54. 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)