LTCC Course Aperiodic Order

Time and Venue

The course runs for five weeks from Monday 27 October 2014 until Monday 24 November 2014. The first lecture will take place in the Burnside room, and following lectures in the Cayley room at De Morgan Hourse from 10:50am until 12:50pm each Monday.

Course Description

This course presents an introduction to the Theory of Aperiodic Order and some of its applications.

symbolic dynamics, discrete geometry, tilings and patterns, inflation tilings, cut and project sets, model sets, almost periodicity, diffraction

Order phenomena are ubiquitous in nature, with crystals being one of the most prominent examples. However, there is no obvious definition of `order', and no complete understanding of what manifestions of `order' may exist. Partly motivated by the discovery of quasicrystals, recognised by the award of the 2011 Nobel Prize in Chemistry to Dan Shechtman, systematic investigations of aperiodically ordered systems have produced a good description of systems that, in some sense, are close to periodic. The course will present an introduction to the theory of Aperiodic Order. It will cover the following main themes.

  1. Preliminaries: point sets, symmetry, lattices and crystals, crystallographic restriction, number theoretic tools, Minkowski embedding
  2. Symbolic dynamics: substitutions, fixed point, hull, properties, well-known examples
  3. Inflation/deflation symmetry: basic construction, examples and properties, repetitivity
  4. Model sets: one-dimensional model sets, cut and project schemes, cyclotomic model sets
  5. Diffraction theory: almost periodic functions, Poisson summation formula, diffraction of crystals, simple example of diffraction of aperiodic structure

Recommended reading:

Additional optional reading:

undergraduate mathematics; in particular, basic notions of number theory, group theory, measure theory

Preliminary reading:

Course Materials

Lecture notes and work sheets will be made available below.

Lecture Notes

Work Sheets and Solutions

End of course examination

Please submit the examination by Monday 15 December, either by email to, or by posting it to my Open University address, which you can find on my home page. Any format is accepted as long as it is readable. If you have any questions or find that you need additional time, please get in touch with me.

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Uwe Grimm
last modified 25 November 2014

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