Journal for Geometry and Graphics, Vol. 4, No. 2, pp. 109-114 (2000)

Five Spheres in Mutual Contact

H. S. M. Coxeter

University of Toronto, Department of Mathematics,
100 St. George Street, Toronto, Ontario, M5S 3G3 Canada

Abstract: Consider five mutually tangent spheres having ${5 \choose 2} = 10$ distinct points of contact. If $O$ is one of these ten points, we obtain by inversion two parallel planes with three ordinary spheres sandwiched between them. Since these three are congruent and mutually tangent, their centres are the vertices of an equilateral triangle. Analogously, if four congruent spheres are mutually tangent, their centres are the vertices of a regular tetrahedron. A fifth sphere, tangent to all these four, may be either a larger sphere enveloping them or a small one in the middle of the tetrahedral cluster. In this article it will be shown that here are fifteen spheres, each passing through six of the ten points of contact of the five mutually tangent spheres.

Keywords: inversive space, orthogonality, point at infinity, regular octahedron and tetrahedron, and triangular prism

Classification (MSC2000): 51M04; 51B10

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