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.


Wall tiling at the Alhambra, in Granada, Spain, illustrating wallpaper group 333.

Course Notes (Draft)

Section 0: Preview in Pictures
Section 1:_Introduction to Topology
Section 2:_Non-orientable Surfaces
Section 3:_Properties of Spaces
Section 4:_Classification of Surfaces
Section 5:_Euler Characteristic
Section 6:_Curvature
Section 7:_Gauss Bonnet Theorem
Section 8:_Symmetries and Isometries
Section 9:_Combining Isometries
Section 10:_Finding All Types of Isometries
Section 11:_Kaleidoscopes
Section 12:_Rosettes
Section 13:_Finite Figures
Section 14:_Frieze Patterns
Section 15:_Classifying Frieze Patterns
Section 16:_Paper Cutting and Musical Frieze Patterns
Section 17:_Wallpaper Patterns
Section 18:_The Magic Theorem
Section 19:_Drawing Wallpaper Patterns
Section 20:_Orbifolds
Section 21:_Spherical Symmetry Patterns
Section 22:_Frieze Pattern Notation
Section 23:_Hyperbolic Geometry
Section 24:_Hyperbolic Wallpaper Patterns
Section 25:_Maps and Graphs
Section 26:_Knots
Section 27:_Fractals

Final Project Ideas

Syllabus (Draft)

Draft Syllabus for Math 70