Topology and Symmetry

Course Description (Math 70)

Note: For Spring 2021, students can choose between taking this course fully online or in person.

In this first year seminar, students will explore ideas from topology and geometry and their application to symmetry patterns.

We will start by building intuition for properties of surfaces with games and visualization exercises. We will develop tools to distinguish surfaces and prove impossibility theorems. We will study the curvature of surfaces, including surfaces we find in our kitchens, like cabbage leaves and banana peels.

The focus of the course will then shift to symmetry and the identification of repeating patterns in the world around us, from snowflakes, to frieze patterns on campus buildings, to designs on tapestries and wallpaper, to paintings like those of M.C. Escher. We will relate symmetry patterns to their folded-up counterparts, called orbifolds, and use tools from geometry and topology to determine which patterns are possible and which can never be achieved. We will extend our analysis to spherical and hyperbolic patterns, uncovering some of the shocking differences between Euclidean and non-Euclidean geometry.

Course assignments will include readings, mathematical investigations, design projects such as virtual and physical kaleidoscopes, quizzes, and a final project. The final project will allow students to pursue a theoretical topic (e.g. hyperbolic tilings or map projections), an application (e.g. quasicrystals or patterns on neckties), or a maker project (e.g. 3-dimensional pattern kaleidoscopes or hyperbolic quilts). No prerequisite knowledge is needed.