On The Number of Cylinders Touching a Sphere
Type of DegreePhD Dissertation
DepartmentMathematics and Statistics
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The kissing number problem is a packing problem in geometry where one has to fi nd the maximum number of congruent non-overlapping copies of a given body so that they can be arranged each touching a common copy. The most studied version of this problem is about the kissing number of the unit ball. A similar question was proposed by Wlodzimierz Kuperberg in 1990. Kuperberg asked for the maximum number of non-overlapping in finitely long unit cylinders touching a unit ball. He conjectured that at most six disjoint infi nitely long unit cylinders can touch a unit sphere. W. Kuperberg's so called six cylinder problem [WK90] is a well known, 28 year old problem in discrete geometry and it is still an open problem. In 2015, Moritz Firsching showed an arrangement of 6 disjoint cylinders with radii 1.0496594, where each cylinder touched a given unit ball. In this dissertation several variants of W. Kuperberg's problem are considered and solved. For example new bounds will be proved concerning the number of tangent cylinders with various radii. Some already known bounds will be improved by elaborating on the method introduced by Brass and Wenk [BW00]. Application of a deep theorem on circle packing by Musin [OM03] also provides some non-trivial bounds. The major part of the dissertation is about proving theorems concerning the size and the number of discs which one can place on a concentric sphere avoiding the cylinders. This way new lower bounds are proved for the total area between cylinders on a concentric sphere. Such lower bounds can improve the existing results concerning Kuperberg's type cylinder problems. Most of the lemmas will be proved with pure geometric arguments, but in some cases the final answer uses Maple computations. We give several different lower bounds for the total area of gaps. Even our best lower bound does not solve Kuperberg's 6 cylinder problem. The last section contains an application of our lower bound (joint work with Andras Bezdek) where it is proved that seven in finitely long cylinders of radii 1.04965 (Firsching's radius) cannot touch a unit sphere. In view of Firsching's construction this settles the Kuperberg question for radius 1.04965.