I. Geometric Transformations

For over two thousand years Euclidean geometry remained the synthetic geometry of Euclid's Elements; lengths and angles were not measured but compared. Then, Rene Descartes (1596–1650) introduced the method of doing geometry using algebra and numbers – analytic geometry. Later, Felix Klein (1849–1925) introduced a more general method (transformations), still involving algebra, that was applicable to other geometries as well. The fundamental geometrical notion of congruence has to do with moving (transforming) a figure to coincide with another. The course studies geometric transformations that preserve distance in the Euclidean plane. These transformations will be seen to describe ALL the symmetry patterns in the plane (and 3‑D which we don't cover). The course begins with an introduction to Analytic geometry which is also useful for Calculus.

II. Non‑Euclidean geometry

Although Non‑Euclidean geometry means geometries other than Euclidean geometry, the reference in mathematics is to two particular geometries – Elliptic geometry and Hyperbolic geometry. The impression of school students and the public in general is that geometry consists of Euclidean geometry, whereas Euclidean geometry is only one of many geometries. However, there are two geometries which are related to Euclidean geometry and whose study gives a greater understanding of what Euclidean geometry is really about. There are three geometries called constant curvature geometries where a 'plane' of the geometry has a curvature that does not change. In Euclidean geometry, the plane is flat in the sense that its curvature is zero. In contrast, a hyperbolic plane has negative curvature like that of a horse saddle and an elliptic plane has positive curvature like a sphere. While the emphasis in this course is these two geometries, we begin with those concepts of Euclidean geometry in which it differs — from the other two constant‑curvature geometries — and the study leads us to Elliptic and Hyperbolic geometry. This course is ideal for building in the student an appreciation of the idea of proof in mathematics and to improve the student's ability to construct proofs.

III. Affine and Projective geometries

The goal of this course is to develop a clear grasp of the various geometries and their relationships to one another. For this reason, we study projective geometry for it entails a unified view and closely relates the different geometries, including Euclidean geometry and non‑Euclidean geometries, as special cases. And the geometry most closely related to projective geometry is affine geometry; studying affine geometry enables greater understanding of projective geometry; so we introduce the former first. A way to see the various geometries, without needing prerequisites in advanced algebra, is the axiomatic approach which we use in the course.

IV. Symmetry and the Idea of a Group

What is symmetry, and how can we describe it mathematically? Beginning with the rigid motions that carry a square onto itself, this course explores the symmetries of plane figures, repeating patterns, and the Platonic solids. Along the way, students discover the idea of a group—a structure that captures how symmetries combine.

We begin by finding and combining the rotations and reflections of a square. Does the order of two transformations matter? Can a few transformations generate all the others? How can we prove that our list of symmetries is complete? These questions lead naturally to composition, identity, inverses, and the group axioms. Our investigations then extend to other figures and to repeating designs. Students explore strip patterns, which repeat along a band, and wallpaper patterns, which repeat across the plane. Translations, rotations, reflections, and glide reflections provide the tools for identifying, comparing, and creating patterns with prescribed symmetries. Moving into three dimensions, we investigate the Platonic solids: the tetrahedron, cube, octahedron, dodecahedron, and icosahedron. Students examine their rotational and reflection symmetries and explore relationships among their vertices, edges, and faces.

Thus, this course develops the group concept through a sustained mathematical investigation of symmetry. By exploring rigid motions of figures and solids, strip patterns, and wallpaper patterns, students discover how groups arise naturally and provide a common language for their observations. We encourage students subsequently to take a standard course in group theory at school or university, where they can develop the abstract theory more systematically, appreciate its broader reach, and recognize symmetry as a central example that gives the theory meaning.

V. Set Theory and Beyond

This course, Axiomatic Set Theory, explores the foundational principles of set theory using an axiomatic approach, focusing on the concepts of well-ordered sets and the hierarchy of infinities. Students will learn the wonderful constructions of the natural numbers, rationals and real numbers out of the empty set and the axioms, understand the properties of infinite sets, and delve into the fascinating realm of transfinite numbers.