Problems in Discrete and Continuous Geometry

Abstract

This dissertation summarizes our work in various problems that are inherently geometric or geometrically motivated. The unifying theme of these problems is that they are easily stated but rife with sizable mathematical complications tied to various different areas. In Chapter 1, we consider the decomposition of graphs into a union of regular graphs. Chapter 2 explores variants of a dynamical system that stabilizes into a set of concentric circles. In Chapter 3, we study the geometry of certain matrices and how it acts on the \(\{\pm 1\}^n\) cube. Chapter 4 serves as a collection of smaller scale projects that consist of investigating the Hausdorff dimension of a one-parameter family of iterated function systems and the similarity of curves to their evolutes and evolutoids. In Chapter 5, we explore the combinatorial game Juniper Green and its connection to the Gallai-Edmonds Decomposition. We conclude in Chapter 6 with a numerical evidence for a conjecture that a stellated tetrahedron is not Rupert.