**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.

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

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