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## On The Number of Cylinders Touching a Sphere

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##### Date

2019-06-14##### Author

Yardimci, Osman

##### Type of Degree

PhD Dissertation##### Department

Mathematics and Statistics##### Restriction Status

EMBARGOED##### Restriction Type

Full##### Date Available

05-04-2021##### Metadata

Show full item record##### Abstract

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.

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