Geodesic
Five Mutually Tangent Spheres and Various Associated Configurations
Canadian mathematical bulletin, Tome 25 (1982) no. 2, pp. 149-163

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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.
DOI : 10.4153/CMB-1982-021-7
Mots-clés : 51M99, Inversive plane, Configurations of spheres, circles and points, Pascal and Cremona configurations, Money-Coutts configuration, Loxodromic sequences of tangent spheres 
Rigby, J. F. Five Mutually Tangent Spheres and Various Associated Configurations. Canadian mathematical bulletin, Tome 25 (1982) no. 2, pp. 149-163. doi: 10.4153/CMB-1982-021-7
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