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Tytuł artykułu

On rational radii coin representations of the wheel graph

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Abstrakty
EN
A flower is a coin graph representation of the wheel graph. A petal of a flower is an outer coin connected to the center coin. The results of this paper are twofold. First we derive a parametrization of all the rational (and hence integer) radii coins of the 3-petal flower, also known as Apollonian circles or Soddy circles. Secondly we consider a general n-petal flower and show there is a unique irreducible polynomial Pₙ in n variables over the rationals ℚ, the affine variety of which contains the cosinus of the internal angles formed by the center coin and two consecutive petals of the flower. In that process we also derive a recursion that these irreducible polynomials satisfy.
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Twórcy
  • Department of Mathematical Sciences, George Mason University, Fairfax, VA, USA
  • Mathematics Department, Hood College, Frederick, MD, USA
Bibliografia
  • [1] E.M. Andreev, Convex polyhedra in Lobačevskiĭ spaces, Matematicheskiĭ Sbornik. Novaya Seriya 81 (123) (1970) 445-478.
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  • [5] R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks and C.H. Yan, Apollonian circle packings: number theory, J. Number Theory 100 (2003) 1-45. doi: 10.1016/S0022-314X(03)00015-5
  • [6] R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks and C.H. Yan, Apollonian circle packings: geometry and group theory. I. The Apollonian group, Discrete Comput. Geom. 34 (4) (2005) 547-585. doi: 10.1007/s00454-005-1196-9
  • [7] R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks and C.H. Yan, Apollonian circle packings: geometry and group theory. II. Super-Apollonian group and integral packings, Discrete Comput. Geom. 35 (1) (2006) 1-36. doi: 10.1007/s00454-005-1195-x
  • [8] R.L. Graham, J.C. Lagarias, C.L. Mallows, A.R. Wilks and C.H. Yan, Apollonian circle packings: geometry and group theory. III. Higher dimensions, Discrete Comput. Geom. 35 (1) (2006) 37-72. doi: 10.1007/s00454-005-1197-8
  • [9] E. Fuchs and K. Sanden, Some experiments with integral Apollonian circle packings, Exp. Math. 20 (4) (2011) 380-399. doi: 10.1080/10586458.2011.565255
  • [10] T. Hungerford, Algebra, Graduate Texts in Mathematics, GTM-73 Springer-Verlag, 1974.
  • [11] P. Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs, Akademie der Wissenshaften Leipzig, Math.-Phys. Klasse 88 (1936) 141-164.
  • [12] MAPLE, mathematics software tool for symbolic computation, http://www.maplesoft.com/products/Maple/academic/index.aspx
  • [13] K.H. Rosen, Elementary Number Theory and Its Applications, Pearson Addison Wesley, 2005.
  • [14] K. Stephenson, Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, 2005.
  • [15] W. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press, 1997.
  • [16] G.M. Ziegler, Lectures on Polytopes, Graduate Texts in Mathematics, GMT-152 Springer Verlag, 1995. doi: 10.1007/978-1-4613-8431-1
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Bibliografia
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bwmeta1.element.bwnjournal-article-doi-10_7151_dmal_1200
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