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Beschreibung:
While the tangency graphs of disk packings are completely characterised by Koebe-Andreev-Thurston’s disk packing theorem, not so much is known about the combinatorics of ball packings in higher dimensions. This thesis tries to make a contribution by investigating some higher dimensional ball packings. The key idea throughout this thesis is a correspondence between balls in Euclidean space and space-like directions in Lorentz space. This allows us to interpret a ball packing as a set of points in the projective Lorentz space. Our investigation starts with Descartes’ configuration, the simplest ball packing studied in this thesis. It serves as the basic element for further constructions. Then we explicitly construct some small packings whose tangency graph can be expressed as graph joins, and identify some graph joins that are not the tangency graph of any ball packing. With the help of these examples, we characterise the tangency graphs of Apollonian ball packings in dimension 3 in terms of the 1-skeletons of stacked polytopes. Partial results are obtained for higher dimensions. Boyd-Maxwell packings form a large class of ball packings that are generated by inversions, generalising Apollonian packings. Motivated by their appearance in recent studies on limit roots of infinite Coxeter systems, we revisit Boyd-Maxwell packings. We prove that the set of limit roots is exactly the residual set of Boyd-Maxwell packings. Furthermore, we describe the tangency graph of a Boyd-Maxwell packing in terms of the corresponding Coxeter complex, and complete the enumeration of Coxeter groups generating these packings. We then propose a further generalization, which may exist in much higher dimensions. Motivated by a result of Benjamini and Schramm, we also study ball packings whose tangency graph is a higher dimensional grid graph. We give a loose bound on the size of such grid graphs that admit a ball packing. ; Während die Kontaktgraphen von Kreispackungen vollständig durch den Satz von Koebe-Andreev-Thurston beschrieben sind, ...