PROJECT III 2015-16

Cutting Sequences

Here is a simple example. Imagine the plane filled with a square grid and a straight line L is drawn across it in such a way that it does not pass through any of the corners of the squares. Now record the sequence (the cutting sequence) of intersections of this line with the grid, recording if the intersection is where L cuts a vertical line (v) or a horizontal line (h). If the line is at a rational slope with respect to the grid then this will be a periodic sequence of v`s and h`s. But if it is not, and L lies at an irrational angle, then the resulting sequence is much more interesting.

The resulting maths analyzing sequences such as this (and related constructions) has powerful links with combinatorics, surfaces and their foliations, number theory, continued fractions, dynamics, topology and mathematical billiards. This project can be taken in a number of ways and will explore both examples and theory related to these ideas.

PREREQUISITES AND COREQUISITES

Basic background in Level 2 pure maths modules would give sufficient prerequisites to study this project.

RESOURCES

There are many good resources on the web. Searching for basic articles mathematical billiards on Wolfram Mathworld, or Sturmian words on Wikipedia would certainly give some good introductions.

EMAIL

John Hunton (mailto:john.hunton@durham.ac.uk)