![]() ![]() In this paper we introduce and study the class of d-ball packings arising from edge-scribable polytopes. ![]() In passing, we give a sufficient condition for a Coxeter graph to generate mutually tangent spheres, and use this to identify an Apollonian sphere packing in three dimensions that is not the Soddy sphere packing. The packings in seven and eight dimensions are different than those found in an earlier paper. Maxwell described all three packings but seemed unaware that they are Apollonian. We use the packing to generate an Apollonian packing in nine dimensions, and a cross section in seven dimensions that is weakly Apollonian. The $E_7$, $E_8$ and Reye lattices play roles. In this paper, we describe an Apollonian packing in eight dimensions that naturally arises from the study of generic nodal Enriques surfaces. We call a packing of hyperspheres in $n$ dimensions an Apollonian sphere packing if the spheres intersect tangentially or not at all they fill the $n$-dimensional space and every sphere in the packing is a member of a cluster of $n+2$ mutually tangent spheres (and a few more properties described herein). In the second method, we use the fractal structure of orthoplicialĪpollonian packings to construct necklace representations of rational links with Number of spheres needed to construct a necklace representation in terms of theĬrossing number. Thurston Circle Packing Theorem and gives a linear upper bound on the minimum The first method follows directly from the Koebe-Andreev. This thesis, we introduce two methods of construction of necklace representations Of Apollonian packings into a novel direction in the area of topology. The polytopal approach that we propose also allows us to extend the applications Or cubical Apollonian packing is contained in the set of curvatures of an integral Then, we introduce the notion of Apollonian section,Īnd we use it to show that the set of curvatures of any integral tetrahedral, octahedral Of the Descartes’ Theorem to find integrality conditions of the Apollonian packingsīased on the Platonic solids. On the regular polytopes in every dimension. Generalization of the Descartes’ Theorem for the sphere packings which are based The polytopal structure also allows us to obtain a This connection, Apollonian packings can be generalized in different geometric settingsĪnd in higher dimensions. Packings where the combinatorics is carried by an edge-scribed polytope. In this thesis, we study a class of sphere Their applications in number theory, geometric group theory, hyperbolic geometry,įractal structures and discrete geometry. In the last decades, Apollonian packings have drawn increasing attention due to ![]()
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