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Journal of Geometry

, Volume 104, Issue 3, pp 421–438 | Cite as

An interstice relationship for flowers with four petals

  • Steve Butler
  • Ron Graham
  • Gerhard Guettler
  • Colin Mallows
Article
  • 94 Downloads

Abstract

Given three mutually tangent circles with bends (related to the reciprocal of the radius) a, b and c respectively, an important quantity associated with the triple is the value \({\langle a,b,c \rangle:=ab+ac+bc}\) . In this note we show in the case when a central circle with bend b 0 is “surrounded” by four circles, i.e., a flower with four petals, with bends b 1, b 2, b 3,b 4 that either
$$\sqrt{\langle b_{0},b_{1},b_{2} \rangle}+\sqrt{\langle b_{0},b_{3},b_{4} \rangle}=\sqrt{\langle b_{0},b_{2},b_{3} \rangle}+\sqrt{\langle b_{0},b_{4},b_{1} \rangle}$$
or
$$\sqrt{\langle b_{0},b_{1},b_{2} \rangle}=\sqrt{\langle b_{0},b_{2},b_{3} \rangle}+\sqrt{\langle b_{0},b_{3},b_{4} \rangle}+\sqrt{\langle b_{0},b_{4},b_{1} \rangle}$$
(where \({\langle b_{0},b_{1},b_{2} \rangle}\) is chosen to be maximal). As an application we give a sufficient condition for the alternating sum of the \({\sqrt{\langle a,b,c\rangle}}\) of a packing in standard position to be 0. (A packing is in standard position when we have two circles with bend 0, i.e., parallel lines, and the remaining circles are packed in between.)

Mathematics Subject Classification (2010)

52C26 

Keywords

Interstice apollonian petals flowers 

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Copyright information

© Springer Basel 2013

Authors and Affiliations

  • Steve Butler
    • 1
  • Ron Graham
    • 2
  • Gerhard Guettler
    • 3
  • Colin Mallows
    • 4
  1. 1.Iowa State UniversityAmesUSA
  2. 2.UC San DiegoLa JollaUSA
  3. 3.University of Applied Sciences Giessen FriedbergGiessenGermany
  4. 4.Avaya LabsBasking RidgeUSA

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