Full Stream Name: Geometry, Symmetry, Groups and Fields

Principle Investigator: Michael Starbird

Research Educator: Mark Daniels

Credit Options: Spring & Fall

Spring: M325K Discrete Mathematics- FRI

Summer: Directed Reading Program (informal)

Fall: Directed Reading Program and advising into relevant mathematics classes

This stream will explore the following questions:

What are transformations in the plane?
What is an isometry?
How can we classify all the possible symmetries of plane figures?

What sorts of combinations (groups) of symmetries can we see in a given plane figure? (In answering this question, we are introduced to permutations, the symmetry groups, and the finite symmetric and dihedral groups.)

What happens when we combine (compose) two or more symmetries?

What sets of symmetries can be used to generate all the symmetries of the plane?

What is the difference between groups, rings, and fields as algebraic structures?

While exploring these questions, students will quickly run into fundamental questions about what it means to do mathematics: how can we define the terms we are using to make sure everyone has the same understanding? How do you convincingly show that you know something must be true, rather than just seeing that it’s true for some specially chosen examples? How do we communicate mathematical ideas to others?