# On rational radii coin representations of the wheel graph

• Volume: 33, Issue: 2, page 167-199
• ISSN: 1509-9415

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## Abstract

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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.

## How to cite

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Geir Agnarsson, and Jill Bigley Dunham. "On rational radii coin representations of the wheel graph." Discussiones Mathematicae - General Algebra and Applications 33.2 (2013): 167-199. <http://eudml.org/doc/270607>.

abstract = {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.},
author = {Geir Agnarsson, Jill Bigley Dunham},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {planar graph; coin graph; flower; polynomial ring; Galois theory},
language = {eng},
number = {2},
pages = {167-199},
title = {On rational radii coin representations of the wheel graph},
url = {http://eudml.org/doc/270607},
volume = {33},
year = {2013},
}

TY - JOUR
AU - Jill Bigley Dunham
TI - On rational radii coin representations of the wheel graph
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2013
VL - 33
IS - 2
SP - 167
EP - 199
AB - 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.
LA - eng
KW - planar graph; coin graph; flower; polynomial ring; Galois theory
UR - http://eudml.org/doc/270607
ER -

## References

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6. [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 Zbl1085.52010
7. [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 Zbl1085.52011
8. [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 Zbl1085.52012
9. [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 Zbl1259.11065
10. [10] T. Hungerford, Algebra, Graduate Texts in Mathematics, GTM-73 Springer-Verlag, 1974.
11. [11] P. Koebe, Kontaktprobleme der konformen Abbildung, Ber. Verh. Sächs, Akademie der Wissenshaften Leipzig, Math.-Phys. Klasse 88 (1936) 141-164.
12. [12] MAPLE, mathematics software tool for symbolic computation, http://www.maplesoft.com/products/Maple/academic/index.aspx
13. [13] K.H. Rosen, Elementary Number Theory and Its Applications, Pearson Addison Wesley, 2005.
14. [14] K. Stephenson, Introduction to Circle Packing: The Theory of Discrete Analytic Functions, Cambridge University Press, 2005. Zbl1074.52008
15. [15] W. Thurston, Three-Dimensional Geometry and Topology, Princeton University Press, 1997. Zbl0873.57001
16. [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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