Abstract: A fundamental problem in distance geometry aims to recover a finite tuple of points, viewed up to oriented isometry, from a small collection of input measurements. This thesis explores approaches to this problem based on the use of frieze patterns, numerical arrays whose entries satisfy certain local algebraic relations.

The thesis consists of two main parts. The first part focuses on quadratic 3-term relations that underlie Coxeter-Conway frieze patterns. It surveys and extends existing work interpreting the values appearing in these relations as geometric measurement data, and establishes direct connections between several geometric contexts in which these relations arise.

The aim of the second part of the thesis is to exhibit the broader applicability of frieze patterns as a tool in distance geometry. We identify measurement data that determines a finite configuration of points on a two-dimensional sphere in three-dimensional Euclidean space. Extending the work of Fomin and Setiabrata, we introduce spherical Heronian and Cayley-Menger frieze patterns that organize this measurement data. Like classical Coxeter-Conway frieze patterns, these new frieze patterns exhibit glide symmetry and a form of the Laurent phenomenon.