Richard Evan Schwartz

Publications

Preprints

• The Four Color Theorem meets Shapes of Polyhedra
  preprint, 2026 pdf
  companion program
  
  
• Mostow Rigidity Made Easier
  preprint, 2025 pdf
  HTML (arXiv) version
  
  
• 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
  
  
• Patterns of Geodesics, Shearing, and Anosov Representations of the Modular Group
  preprint (2025) pdf
  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

• Build Boy's Surface
  AMR survey
  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