2018 LTCC Course Aperiodic Order

Time and Venue

The course runs for five weeks from Monday 12 November 2018 until Monday 10 December 2018. The lectures will take place in the Burnside room at De Morgan House starting at 3:10pm each Monday.

Course Description

This course will present an introduction to the mathematical theory of structures such as sequences, point sets, tilings and patterns that exhibit non-periodic order. A prominent example is Penrose's fivefold tiling. This topic combines elements from discrete geometry, (symbolic) dynamics, number theory and harmonic analysis with applications in crystallography.


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

Course Materials

Lecture notes, work sheets and solutions will be made available below. Note that lecture notes will be updated to contain proofs following the lecture.

Solutions to work sheets will be posted before the following lecture.

Lecture Notes

Work Sheets and Solutions

End of course examination

Please submit the examination by 25 January 2019, either by email to uwe.grimm@open.ac.uk, 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
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