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Polycyclic Movable 4-Configurations are Plentiful

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Abstract

A geometric 4-configuration is a collection of points and straight lines with the property that every point lies on exactly four lines in the collection and every line passes through exactly four points in the collection. This paper describes a method for constructing a large number of new infinite families of rotationally symmetric geometric 4-configurations which are movable; that is, there is at least one continuous parameter which preserves the symmetry of the configuration. In fact, the configurations in this paper have 2q continuous parameters for any integer \(q \ge 2\); previously the known classes of movable 4-configurations had only one or two degrees of freedom. The construction is extended to produce movable 4-configurations with dihedral symmetry. The paper ends with a number of open questions.

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References

  1. Barthel, G., Hirzebruch, F., Höfer, T.: Geradenkonfigurationen und Algebraische Flächen. Aspects of Mathematics, D4. Friedr. Vieweg & Sohn, Braunschweig (1987)

  2. Berardinelli, A., Berman, L.W.: Systematic celestial configurations. Ars Math. Contemp. 7(2), 361–377 (2014)

  3. Berman, L.W.: A characterization of astral \((n_4)\) configurations. Discrete Comput. Geom. 26(4), 603–612 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berman, L.W.: Movable \((n_4)\) configurations. Electron. J. Comb. 13(1), Research Paper 104, 30 (2006)

  5. Berman, L.W.: A new class of movable \((n_4)\) configurations. Ars Math. Contemp. 1(1), 44–50 (2008)

    MathSciNet  MATH  Google Scholar 

  6. Berman, L.W.: Geometric constructions for 3-configurations with non-trivial geometric symmetry. Electron. J. Comb. 20(3), Paper 9, 29 (2013)

  7. Berman, L.W., Faudree, J.R.: Highly incident configurations with chiral symmetry. Discrete Comput. Geom. 49(3), 671–694 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  8. Boben, M., Pisanski, T.: Polycyclic configurations. Eur. J. Comb. 24(4), 431–457 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bokowski, J., Grünbaum, B., Schewe, L.: Topological configurations \((n_4)\) exist for all \(n\ge 17\). Eur. J. Comb. 30(8), 1778–1785 (2009)

    Article  MATH  Google Scholar 

  10. Bokowski, J., Pilaud, V.: Enumerating topological \((n_k)\)-configurations. Comput. Geom. 47(2, part A), 175–186 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Bokowski, J., Schewe, L.: There are no realizable \(15_4\)- and \(16_4\)-configurations. Rev. Roum. Math. Pures Appl. 50(5–6), 483–493 (2005)

    MathSciNet  MATH  Google Scholar 

  12. Bokowski, J., Schewe, L.: On the finite set of missing geometric configurations \((n_4)\). Comput. Geom. 46(5), 532–540 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. Coxeter, H.S.M.: Self-dual configurations and regular graphs. Bull. Am. Math. Soc. 56, 413–455 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  14. Gévay, G.: Constructions for large spatial point-line \((n_k)\) configurations. Ars Math. Contemp. 7(1), 175–199 (2014)

    MathSciNet  MATH  Google Scholar 

  15. Gross, J.L., Tucker, T.W.: Topological Graph Theory. Wiley-Interscience Series in Discrete Mathematics and Optimization. Wiley, New York (1987)

    MATH  Google Scholar 

  16. Grünbaum, B.: Which \((n_4)\) configurations exist? Geombinatorics 9(4), 164–169 (2000)

    MathSciNet  MATH  Google Scholar 

  17. Grünbaum, B.: Connected \((n_4)\) configurations exist for almost all \(n\)—an update. Geombinatorics 12(1), 15–23 (2002)

    MathSciNet  MATH  Google Scholar 

  18. Grünbaum, B.: Connected \((n_4)\) configurations exist for almost all \(n\)—second update. Geombinatorics 16(2), 254–261 (2006)

    MathSciNet  Google Scholar 

  19. Grünbaum, B.: Musings on an example of Danzer’s. Eur. J. Comb. 29(8), 1910–1918 (2008)

    Article  MATH  Google Scholar 

  20. Grünbaum, B.: A catalogue of simplicial arrangements in the real projective plane. Ars Math. Contemp. 2(1), 1–25 (2009)

    MathSciNet  MATH  Google Scholar 

  21. Grünbaum, B.: Configurations of Points and Lines, Graduate Studies in Mathematics, vol. 103. American Mathematical Society, Providence (2009)

    Google Scholar 

  22. Grünbaum, B., Rigby, J.F.: The real configuration \((214)\). J. Lond. Math. Soc. (2) 41(2), 336–346 (1990)

    Article  MATH  Google Scholar 

  23. Pisanski, T., Servatius, B.: Configurations from a Graphical Viewpoint. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser, Boston (2013)

  24. Zagier, D.: Life and work of Friedrich Hirzebruch. Jahresber. Dtsch. Math.-Ver. 117(2), 93–132 (2015)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Leah Wrenn Berman thanks Jürgen Bokowski, at whose Workshop on Configurations the first ladder configuration was found. L. Berman’s research supported by a Grant from the Simons Foundation (#209161 to L. Berman); T. Pisanski’s research supported in part by the ARSS of Slovenia, Research Grants P1-0294, P1-0297 and N1-0011:GReGAS, and the European Science Foundation.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University of Alaska Fairbanks, Fairbanks, AK, 99775-6660, USA

    Leah Wrenn Berman & Jill R. Faudree

  2. University of Ljubljana, Ljubljana, Slovenia

    Tomaž Pisanski

  3. University of Primorska, Koper, Slovenia

    Tomaž Pisanski

Authors
  1. Leah Wrenn Berman
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  2. Jill R. Faudree
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  3. Tomaž Pisanski
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Correspondence to Jill R. Faudree.

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Berman, L.W., Faudree, J.R. & Pisanski, T. Polycyclic Movable 4-Configurations are Plentiful. Discrete Comput Geom 55, 688–714 (2016). https://doi.org/10.1007/s00454-015-9749-z

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  • DOI: https://doi.org/10.1007/s00454-015-9749-z

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