Tilings

Description: Tilings appear as decorative elements everywhere through the whole history of human beings. There are several ways to organise and construct them. We will look at both periodic and aperiodic tilings and will discuss the methods to construct them.

Group project:

We will look at verious types of tilings, methods to construct them and ways to prove that cirtain constructions produce tilings. We will discuss the following questions:

• What is the difference between periodic and aperiodic tilings?
• How to use group actions to construct tilings?
• How to produce many examples of tilings on the sphere, Euclidean plane and hyperbolic plane?
• How to build some aperiodic tilings?

Mode of operation and evidence of learning:
The project will involve learning through reading, with a focus on underlying theory, mathematical rigour, and development of conceptual understanding. Students will work within their group, demonstrating understanding by exploring results and examples, solving relevant problems, and clearly communicating in both written and oral formats.

Individual Project:

The individual project will build on knowledge gained in the group project and explore in depth additional advanced topics. Some examples of topics are:

• Tilings of the sphere based on regular polytopes;
• Reflection groups on the sphere, on Euclidean plane and on hyperbolic plane;
• Higher dimensional tilings;
• Poincare Theorem;
• Aperiodic tilings: Penrose;
• Aperiodic tilings: monotile;
• Applications of tilings to solving geometric problems (Pythagorean theorem, Pizza Theorem, Scissors congruence)

Mode of Operation and Evidence of Learning
The project will involve learning through reading, with a focus on underlying theory, mathematical rigour, and development of conceptual understanding. Students will work individually, demonstrating understanding by exploring results and examples, solving relevant problems, and clearly communicating in both written and oral formats.

Prerequisites and Corequisites:

Algebra II is an essential prerequisite.

Personal interest in geometric and visual aspects of mathematics would help while taking this project.
Whilst not essential corequisites, takings Topology III and/or Galois Theory, Groups and Geometry III could be indication of presence of this interest.

Resources:

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