Mathematics Colloquia and Seminars

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Classical Apollonian circle packings are constructed from a quadruple of pairwise tangent circles on a plane by successively inscribing circles into the triangular interstices. We consider certain variations of this construction, and study a family of sphere packings, which we call "orthoplicial" Apollonian sphere packings. Remarkably, just as in the classical Apollonian circle packings, there are orthoplicial sphere packings in which the curvatures of constituent spheres are integers, giving rise to fascinating questions on the Diophantine properties of the set of curvatures. We describe these packings in terms of hyperbolic geometry, Coxeter groups, and quadratic forms, and discuss the "local-global principle" for the set of curvatures.