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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

IMPACT FACTOR 2017: 0.734

CiteScore 2018: 0.73

SCImago Journal Rank (SJR) 2018: 0.329
Source Normalized Impact per Paper (SNIP) 2018: 0.908

Mathematical Citation Quotient (MCQ) 2017: 0.62

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Volume 18, Issue 2


Negative refraction and tiling billiards

Diana Davis / Kelsey DiPietro
  • Department of Applied and Computational Mathematics and Statistics, University of Notre Dame, 153 Hurley Hall, Notre Dame, IN 46556, USA
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/ Jenny Rustad / Alexander St Laurent
  • Department of Mathematics and Department of Computer Science, Brown University, 151 Thayer Street, Providence, RI 02912, USA
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Published Online: 2018-03-26 | DOI: https://doi.org/10.1515/advgeom-2017-0053


We introduce a new dynamical system that we call tiling billiards, where trajectories refract through planar tilings. This system is motivated by a recent discovery of physical substances with negative indices of refraction. We investigate several special cases where the planar tiling is created by dividing the plane by lines, and we describe the results of computer experiments.

Keywords: Negative refraction; billiards

MSC 2010: 37D50


  • [1]

    W. Barker, R. Howe, Continuous symmetry. Amer. Math. Soc. 2007. MR2362745 Zbl 1131.51001Google Scholar

  • [2]

    D. Dolgopyat, B. Fayad, Unbounded orbits for semicircular outer billiard. Ann. Henri Poincaré 10 (2009), 357–375. MR2511890 Zbl 05843997CrossrefGoogle Scholar

  • [3]

    K. Engelman, A. Kimball, Negative Snell’s propagation. Unpublished, ICERM student presentation archive (2012), http://icerm.brown.edu/html/programs/summer/summer_2012/includes/snell.pdf

  • [4]

    G. A. Galperin, Nonperiodic and not everywhere dense billiard trajectories in convex polygons and polyhedrons. Comm. Math. Phys. 91 (1983), 187–211. MR723547 Zbl 0529.70001CrossrefGoogle Scholar

  • [5]

    D. I. Genin, Regular and chaotic dynamics of outer billiards. PhD thesis, Pennsylvania State University (2005).Google Scholar

  • [6]

    P. Glendinning, Geometry of refractions and reflections through a biperiodic medium. SIAM J. Appl. Math. 76 (2016), 1219–1238. MR3519148 Zbl 1362.37082CrossrefWeb of ScienceGoogle Scholar

  • [7]

    B. Grünbaum, Arrangements and spreads. Amer. Math. Soc. 1972. MR0307027 Zbl 0249.50011Google Scholar

  • [8]

    A. Kwan, J. Dudley, E. Lantz, Who really discovered Snell’s law? Physics World, p. 62, 2002.Google Scholar

  • [9]

    A. Mascarenhas, B. Fluegel, Antisymmetry and the breakdown of Bloch’s theorem for light. Preprint.Google Scholar

  • [10]

    H. Masur, Closed trajectories for quadratic differentials with an application to billiards. Duke Math. J. 53 (1986), 307–314. MR850537 Zbl 0616.30044Google Scholar

  • [11]

    C. McMullen, “Trapped”. GIF image: www.math.harvard.edu/~ctm/gallery/billiards/trapped.gif, accessed 9 June 2015.Google Scholar

  • [12]

    R. E. Schwartz, Obtuse triangular billiards. II. One hundred degrees worth of periodic trajectories. Experiment. Math. 18 (2009), 137–171. MR2549685 Zbl 05644745CrossrefGoogle Scholar

  • [13]

    R. E. Schwartz, Outer billiards on kites, volume 171 of Annals of Mathematics Studies. Princeton Univ. Press 2009. MR2562898 Zbl 1205.37001Google Scholar

  • [14]

    R. E. Schwartz, Outer billiards on the Penrose kite: compactification and renormalization. J. Mod. Dyn. 5 (2011), 473–581. MR2854095 Zbl 1277.37070Web of ScienceCrossrefGoogle Scholar

  • [15]

    R. A. Shelby, D. R. Smith, S. Schultz, Experimental Verification of a Negative Index of Refraction. Science 292 (2001), 77–79.CrossrefGoogle Scholar

  • [16]

    D. Smith, J. Pendry, M. Wiltshire, Metamaterials and negative refractive index. Science 305 (2004), 788–792.CrossrefGoogle Scholar

  • [17]

    S. Tabachnikov, Billiards. Panor. Synth. no. 1 (1995), vi+142. MR1328336 Zbl 0833.58001Google Scholar

  • [18]

    S. Tabachnikov, Geometry and billiards, volume 30 of Student Mathematical Library. Amer. Math. Soc. 2005. MR2168892 Zbl 1119.37001Google Scholar

  • [19]

    S. Tabachnikov, A proof of Culter’s theorem on the existence of periodic orbits in polygonal outer billiards. Geom. Dedicata 129 (2007), 83–87. MR2353984 Zbl 1131.37039CrossrefGoogle Scholar

  • [20]

    G. W. Tokarsky, Galperin’s triangle example. Comm. Math. Phys. 335 (2015), 1211–1213. MR3320310 Zbl 1348.37063CrossrefGoogle Scholar

  • [21]

    F. Vivaldi, A. V. Shaidenko, Global stability of a class of discontinuous dual billiards. Comm. Math. Phys. 110 (1987), 625–640. MR895220 Zbl 0653.58018CrossrefGoogle Scholar

  • [22]

    N. Wolchover, Physicists close in on ‘perfect’ optical lens. Quanta Magazine. 8 Aug 2013.Google Scholar

About the article

Received: 2015-02-07

Revised: 2015-08-31

Accepted: 2016-04-17

Published Online: 2018-03-26

Published in Print: 2018-04-25

Citation Information: Advances in Geometry, Volume 18, Issue 2, Pages 133–159, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2017-0053.

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