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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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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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