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تسجيل مستخدم جديدA polytopal generalization of Apollonian packings and Descartes theorem
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In this paper we introduce and study the class of d-ball packings arising from edge-scribable polytopes. We are able to generalize Apollonian disk packings and the well-known Descartes theorem in different settings and in higher dimensions. After introducing the notion of Lorentzian curvature of a polytope we present an analogue of the Descartes theorem for all regular polytopes in any dimension. The latter yields to nice curvature relations which we use to construct integral Apollonian packings based on the Platonic solids. We show that there are integral Apollonian packings based on the tetrahedra, cube and dodecahedra containing the sequences of perfect squares. We also study the duality, unicity under Mobius transformations as well as generalizations of the Apollonian groups. We show that these groups are hyperbolic Coxeter groups admitting an explicit matrix representation. An unexpected invariant, that we call Mobius spectra, associated to Mobius unique polytopes is also discussed.
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