Project III 2026-2027

The Einstein

Andrew Lobb

Description

When you tile your bathroom, you generally use square tiles; it's also fairly common to use rectangular tiles. It's a little kooky to use triangular or hexagonal tiles, but sometimes you see this. Only the truly cool or sadly nerdy will consider tiling their bathrooms aperiodically.

Recently the 'Einstein' tile was discovered -- a single shape of tile that can tile the infinite plane but in such a way that the pattern never repeats. The Einstein tiling is an example of an aperiodic tiling (up until now such tilings have only been found by using more than one shape of tile). Aperiodic tilings exhibit some regularities, rather than being completely random. In this project we shall examine various aspects of the creation and the mathematics of aperiodic tiling.

Group project

The group project will take as its text the book Tilings and Patterns by Gruenbaum and Shephard.

By the end of the group project you shall have learnt about the basic notions and definitions relevant to tilings, you will have studied tilings by regular polygons, and various generalizations of this.

Individual project

My intention in the individual project is that you will learn one or more aspects of periodic or aperiodic tilings. For example, the topology of these tilings, particular tilings by Wang tiles or by the Einstein, tilings of other surfaces, coloured tilings, and so on. What you choose will be informed by your tastes in the subject.

Note: I intend to be away from Durham for much of Epiphany term, during these weeks the supervision meetings will take place online.

Mode of operation and evidence of learning

Both the group and the individual project will involve learning through reading, with a focus on underlying theory, mathematical rigour, and development of conceptual understanding. Students will demonstrate understanding by exploring results and examples, solving relevant problems, and clearly communicating in both written and oral formats.

Prerequisites

Algebra II is a prerequisite. This project may pair well with Analysis and Topology III, and Codes and Knots III, but neither is necessary.

Preliminary Resources

email: Andrew Lobb.

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