Olli Pettersson: The Geometry of a Good Cut: Moser, Pizza, and Beyond

Abstract

How many regions can be formed by connecting points on a circle with straight lines? What appears at first to be a simple geometric exercise quickly unfolds into a rich interplay between combinatorics, geometry, and topology. This thesis explores a family of classic problems involving partitions of space—starting with Moser’s circle problem, extending through inductive reasoning, binomial identities, Euler’s characteristic, and considerations of higher-dimensional analogues. Along the way, we uncover the seductive illusion of exponential patterns, harness the structure of Pascal’s triangle, and reflect on how different mathematical perspectives—combinatorial, visual, and topological—can complement one another. A variety of approaches are explored to shed light on the creative and multifaceted nature of mathematics—qualities that often stand in contrast to how the subject is presented in traditional curricula, where time and space for multiple methods are rare. This work aims to celebrate mathematical thinking as an art of variation and discovery.