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Five Mutually Tangent Spheres and Various Associated Configurations

Published online by Cambridge University Press:  20 November 2018

J. F. Rigby
J. F. Rigby*
Affiliation:
University CollegeCardiff Cf1 1XL, Wales
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Abstract

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If five spheres σ0, σ1, …, σ4 touch each other externally and have radii in geometrical progression, there is a dilative rotation mapping σ0, σ1, σ2, σ3, to σ1, σ2, σ3, σ4; the dilatation factor is shown to be negative. The ten points of contact of the spheres lie by fours on 15 circles, forming a (154106) configuration in inversive space. In the corresponding configuration in the inversive plane, the 15 circles meet again in 60 points, which lie by fours on 45 circles touching by threes at each of the 60 points, and forming a configuration isomorphic to that of 60 Pascal lines (associated with six points on a conic) meeting by fours at 45 points. The 45 circles arise from ten Money-Coutts configurations of nine anti-tangent cycles. Conjectures are made about other circles through the 60 points.

Keywords

51M99Inversive planeConfigurations of spherescircles and pointsPascal and Cremona configurationsMoney-Coutts configurationLoxodromic sequences of tangent spheres
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 25 , Issue 2 , 01 June 1982 , pp. 149 - 163
Copyright
Copyright © Canadian Mathematical Society 1982

References

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