Universität zu Köln

Spomer, Andreas ORCID: 0009-0008-3966-7669 (2026). Packing Regular Spherical Polygons. PhD thesis, Universität zu Köln.

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Abstract

This thesis studies packing problems for regular spherical polygons on the two-dimensional unit sphere. It develops techniques for constructing lower and upper bounds for the packing density of spherical polygons with varying sizes and numbers of vertices. A central contribution of this thesis is the introduction of a necessary and sufficient algebraic criterion for deciding when two congruent regular spherical polygons within a prescribed class have disjoint interiors. This characterization serves as the principal tool underlying both the constructive lower bounds and the analytic upper bounds established in the thesis. To obtain lower bounds, we formulate the problem of maximizing packing density for a fixed number of regular spherical polygons as an optimization problem on a Riemannian submanifold. We describe numerical methods for approximating stationary points and show how to convert the resulting data into rigorous configurations. We use three complementary approaches to establish upper bounds. The strongest among them is an extension of the Lovász theta number to Cayley graphs on the special orthogonal group. Its computation is based on the representation theory of the special orthogonal group and leads to optimization problems involving trigonometric sum-ofsquares polynomials. By exploiting the symmetries of regular spherical polygons, we significantly reduce the complexity of these problems.

Item Type:	Thesis (PhD thesis)
Creators:	Creators Email ORCID ORCID Put Code Spomer, Andreas andreas.spomer1996@gmail.com https://orcid.org/0009-0008-3966-7669 UNSPECIFIED
URN:	urn:nbn:de:hbz:38-810372
Date:	2026
Language:	English
Faculty:	Faculty of Mathematics and Natural Sciences
Divisions:	Faculty of Mathematics and Natural Sciences > Department of Mathematics and Computer Science > Mathematical Institute
Subjects:	Mathematics
Uncontrolled Keywords:	Keywords Language Spherical polygon packings English packing problems English spherical geometry English discrete geometry English geometric optimization English trigonometric polynomial optimization English semidefinite optimization English
Date of oral exam:	13 July 2026
Referee:	Name Academic Title Vallentin, Frank Prof. Dr. Schürmann, Achill Prof. Dr.
Refereed:	Yes
URI:	http://kups.ub.uni-koeln.de/id/eprint/81037