Possibly relevant: arxiv.org/pdf/math/0101066 Beyond the Descartes Circle Theorem Jeffrey C. Lagarias Colin L. Mallows Allan R. Wilks AT&T Labs, Florham Park, NJ 07932-0971 (January 8, 2001) ABSTRACT The Descartes circle theorem states that if four circles are mutually tangent in the plane, with disjoint interiors, then their curvatures (or “bends”) b i = 1 r i satisfy the relation (b 1 + b 2 + b 3 + b 4 ) 2 = 2(b 21 + b 2 2 + b 2 3 + b 2 4 ). We show that similar relations hold involving the centers of the four circles in such a configuration, coordinatized as complex numbers, yielding a complex Descartes Theorem. These relations have elegant matrix generalizations to the n-dimensional case, in each of Euclidean, spherical, and hyperbolic geometries. These include analogues of the Descartes circle theorem for spherical and hyperbolic space. AMS Subject Classification (2000): 52C26 (Primary) 11H55, 51M10, 53A35 (Secondary) Keywords: Apollonian packing, circle packings, inversive geometry, hyperbolic geometry, spherical geometry Also R. L. Graham, J. C. Lagarias, C. L. Mallows, A. Wilks and C. Yan, Apollonian Packings: Geometry and Group Theory I. The Apollonian Group, eprint: arXiv math.MG/0010298 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. Wilks and C. Yan, Apollonian Packings: Geometry and Group Theory II. Super- Apollonian Group and Integral Packings, eprint: arXiv math.MG/0010302 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. Wilks and C. Yan, Apollonian Packings: Geometry and Group Theory III. Higher Dimensions, eprint: arXiv math.MG/0010324 R. L. Graham, J. C. Lagarias, C. L. Mallows, A. Wilks and C. Yan, Apollonian Circle Packings: Number Theory, eprint: arXiv math.NT/0009113 At 10:59 AM 7/14/2014, Bill Gosper wrote:

(at the origin) bisects the unit 2-sphere at (1,1,1): Graphics3D[{Sphere[{0, 0, 0}, 2], Sphere[{1, 1, 1}]}, PlotRange -> {{0, 2}, {0, 2}, {0, 2}}] gosper.org/2spheres.png --rwg