Chapter 1 Introduction

In this project, we study integer circles in $\mathbb{Z}^{2}$. Particularly, we study the density of intersections of integer circles and the existence of inscribed circles.

In this project we work within the context of integer geometry. Here, the length of a segment (or the distance between two points) is the number of lattice points that the segment passes through.

For main theory on geometry of numbers, see e.g..

The notion of an integer circle was introduced in . This article studies the notion of circumscribed circles in integer geometry, studying their existence and their radii. The notion of integer circles introduced here gives a geometric interpretation of relatively prime numbers, the density of which was studied is shown to be the sum of the reciprocals of square numbers . This value is $\frac{\pi^{2}}{6}$, and was first solved by Leonard Euler. In this report, we consider the centres of these circumscribed circles. Note that the notion of integer circles relate to the limit of Farey sunbursts, which date back to 1816.

The book "Geometry of Continued Fractions" gives an introduction to the study of integer geometry which we follow here.

The main results of this project are the construction of formulae to find the density of the intersection of integer circles and the bisection of an integer angle. This latter result is a consequence of the inscribed circles of an integer angle.

In Chapter 2 we introduce the basic definitions of integer geometry, densities and computational background for later results. In Chapter 3 we investigate the density of the intersection of integer circles. In Chapter 4 we study the relation between the centres of circumscribed circles and the set itself and in Chapter 5 we find the inscribed circles of a set of lines, half-lines or of a polygon in Integer Geometry.

While working on this project, I developed skills in creating good numerical tests using a computer and learned when these guesses can be useful in finding results in pure mathematics. I learned more about integer geometry and touched on some number theory that I haven’t had experience with outside of this project.