# On rational radii coin representations of the wheel graph

Geir Agnarsson; Jill Bigley Dunham

Discussiones Mathematicae - General Algebra and Applications (2013)

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

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topGeir 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>.

@article{GeirAgnarsson2013,

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 - Geir Agnarsson

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 -

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