Richard Evan Schwartz

Publications

Preprints

• On Pappus and Anosov Representations of the Modular Group
  preprint (2026) pdf
  
  
• Patterns of Geodesics, Shearing, and Anosov Representations of the Modular Group
  (The original paper with this name is now split into two,
  with the above listed paper being the first half)
  preprint (2026) pdf
  
  
• The Four Color Theorem meets Shapes of Polyhedra
  preprint, 2026 pdf
  companion program
  
  
• Monotonicity in Complex Hyperbolic Triangle Groups
  preprint, 2025 pdf
  
  
• Collapsibility and Near Universality for Vertex Minimal Paper Tori
  preprint, 2025 pdf
  companion programs
  (joint with Peter Doyle)
  HTML (arXiv) version
  
  
• Vertex-Minimal Paper Tori
  preprint, 2025 pdf
  companion programs
  supplementary material pdf
  HTML (arXiv) version
  
  
• The Topological Salesman Problem
  preprint, 2025 pdf
  
  
• Divide and Conquer, a Distributed Approach to Five Point Energy Minimization
  the eternal preprint, (revised in 2025) pdf
  companion programs
  HTML (arXiv) version
  
  
• The Crisscross and the cap: Two Short 3-twist Paper Moebius bands
  preprint (2023) pdf (joint with Brienne Brown)
  
  
• The Farthest Point Map on the Regular Dodecahedron
  preprint (2021) pdf
  companion program
  HTML (arXiv) version
  
  
• On Area Growth in Sol
  preprint (2020) pdf
  HTML (arXiv) version
  
  
• Outer Billiards, Polytope Exchange Transformations, and Quarter Turn Compositions
  preprint (2016) pdf
  
  
  
• Outer Billiards, The Arithmetic Graph, and the Octagon
  preprint (2010) pdf
  Companion programs: OctoMap 1 and OctoMap 2
  
  

Monographs

• 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
  
  companion online material
  HTML (arXiv) version
  
  
• Spherical CR Geometry and Dehn Surgery
  Annals of Mathematics Studies 165 (2007)
  the publication

Papers

• Mostow Rigidity Made Easier
  Bulletin of the A.M.S. (2026) to appear pdf
  HTML (arXiv) version
  
  
• Build Boy's Surface
  AMR survey (2026)
  HTML (arXiv) version
  
  
• The Most Efficient Origami Torus
  Proceedings of the Natl. Acad. of Sciences (2026) pdf to appear
  
  
• A Geometric View of a Continuous Nowhere Differentiable Function
  Mathematical Intelligencer (2026) pdf to appear
  
  
• Symplectic Tiling Billiards, Planar Linkages, and Hyperbolic Geometry
  Bulletin of the A.M.S. (2025) pdf to appear
  HTML (arXiv) version
  
  
• The Optimal Twisted Paper Cylinder
  Advances in Geometry (2025) to appear pdf (joint with Noah Montgomery)
  HTML (arXiv) version
  
  
• Le Retour de Pappus
  KIAS - Springer Series in Mathematics (2025) pdf
  HTML (arXiv) version
  
  
• The Flapping Birds in the Pentagram Zoo
  Arnold Math Journal (2025) to appear pdf
  companion program
  HTML (arXiv) version
  
  
• On Nearly Optimal Paper Moebius Bands
  Advances in Geometry (2025) to appear pdf
  HTML (arXiv) version
  
  
• The Optimal Paper Moebius Band
  Annals of Mathematics Vol 201.1 (2025) pp 291-305 pdf
  additional material
  HTML (arXiv) version
  
  
• Pentagram Rigidity for Centrally Symmetric Octagons
  I.M.R.N (2024) to appear pdf
  companion program
  HTML (arXiv) version
  
  
• Continued Fractions and the Four Color Theorem
  Journal of Experimental Math (2024) to appear pdf
  
  
• Polygonal Symplectic Billiards
  (joint with Peter Albers, Guatam Banhatti, Filip Sadlo, Serge Tabachnikov)
  Journal of Experimental Math (2023) to appear
  HTML (arXiv) version
  
  
• 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
  HTML (arXiv) version
  
  
• Trisecting the 9-vertex complex projective plane
  Math. Intelligencer (2022) to appear pdf
  HTML (arXiv) version
  
  
• On Projective Evolutes of Polygons
  Experimental Mathematics (2022) to appear pdf
  companion program
  (Joint with Maxim Arnold and Serge Tabachnikov)
  HTML (arXiv) version
  
  
• Conway's Nightmare: Brahmagupta and Butterflies
  Math. Intelligencer (2022) to appear pdf
  HTML (arXiv) version
  
  
• Survey Lecture on Billiards
  Proceedings of the 2022 ICM (2022) to appear pdf
  
  
• An Improved Bound on the Optimal Embedded Paper Mobius Band
  Geometriae Dedicata (2021)
  Corrected version: pdf
  Erratum for the original version: pdf
  HTML (arXiv) version
  
  
• The Spheres of Sol
  (joint with Matei P. Coiculescu)
  Geometry and Topology (2021) to appear pdf
  Companion program download
  HTML (arXiv) version
  
  
• The Farthest Point Map on the Regular Octahedron
  Journal of Experimental Math (2021) to appear pdf
  Companion program download
  HTML (arXiv) version
  
  
• Descartes Circle Theorem, Steiner Porism, and Spherical Designs
  (joint with Serge Tabachnikov) American Math Monthly Vol 127 Issue 3 (2020)
  HTML (arXiv) version
  
  
• A Trichotomy for Rectangles inscribed in a Jordan Loop
  Geometriae Dedicata (2020) to appear pdf
  HTML (arXiv) version
  
  
• Inscribed Rectangle Coincidences
  Advances in Geometry (2019) to appear. pdf
  HTML (arXiv) version
  
  
• 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
  HTML (arXiv) version
  
  
• Lenghtening a Tetrahedron
  Geometriae Dedicata (to appear) (2014) pdf
  Companion program download
  HTML (arXiv) version
  
  
• 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
  HTML (arXiv) version
  
  
• 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 )
  HTML (arXiv) version
  
  
• The 5-Electron Case of Thomson's Problem
  J. Experimental Math, vol 22, issue 2, (2013) pp 157-186 pdf
  companion program Electron Champ
  HTML (arXiv) version
  
  
• 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
  HTML (arXiv) version
  
  
• The Pentagram Integrals for Inscribed Polygons
  Electronic Journal of Combinatorics (2011) to appear
  pdf or companion program (download)
  joint with Sergei Tabachnikov )
  HTML (arXiv) version
  
  
• Outer Billiards and the Pinwheel Map
  Journal of Modern Dynamics Issue 2 (2011) 255-283 pdf
  Companion Applet: Pinwheel
  HTML (arXiv) version
  
  
• 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
  HTML (arXiv) version
  
  
• Billiards in Nearly Isosceles Triangles
  Journal of Modern Dynamics, Vol 3 Number 2 (2009) pp 159-231 pdf
  (joint with Pat Hooper)
  HTML (arXiv) version
  
  
• 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)
  .pdf
  HTML (arXiv) version
  
  
• 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)
  .pdf
  
  
• Affine Subdivision, Steerable Semigroups, and Sphere Coverings
  Quarterly Journal of Pure and Applied Mathematics
  (Special volume in honor of G.A. Margulis' 60th Birthday.)
  .pdf
  
  
• Obtuse Triangular Billiards I: Near the (2,3,6) Triangle
  Journal of Experimental Mathematics (2006) Vol 15, Issue 2
  .pdf
  
  
• 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
  HTML (arXiv) version
  
  
• Modular Circle Quotients and PL Limit Sets
  Geometry&Topology Volume 8 (2004) Paper 1 pp 1-34
  HTML (arXiv) version
  
  
• The Density of Shapes in Three Dimensional Barycentric Subdivision
  Journal of Discrete and Computational Geometry #30 (2003) 373-377
  .pdf
  
  
• 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
  .pdf
  HTML (arXiv) version
  
  
• Lucy and Lily: a Game of Geometry and Number Theory
  American Math Monthly
  (Jan 2002) Vol 109.1, pp 13-20
  .pdf
  
  
• Circle Quotients and String Art
  Topology (2001) Vol 41.3 pp. 495-523
  .pdf
  
  
• The Pentagram Map is Recurrent
  Journal of Experimental Mathematics
  (2001) Vol 10.4 pp. 519-528
  .pdf
  
  
• Degenerating the Complex Hyperbolic Ideal Triangle Groups
  Acta Mathematica 186
  (2001) Vol 186 p.105-154
  .pdf
  
  
• 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
  .pdf
  
  
• Symmetric Patterns of Geodesics and Automorphisms of Surface Groups
  Inventiones Mathematicae
  (1997) Vol 128 pp. 177-199
  .pdf
  
  
• 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