# Combinatorics of Sequences and Tilings and its Applications

Lead Research Organisation:
The Open University

Department Name: Applied Mathematics

### Abstract

One of the intriguing aspects of Nature is the symmetry and order apparent in the world around us, for instance in the shape of crystals, or, at a microscopic level, in the regular arrangement of atoms making up the crystal. We have a surprisingly limited understanding of the origin of order and symmetry; and, maybe even more surprisingly, no clear mathematical definition of the concept of order exists. In the example of the crystal, the underlying order is apparent in the periodic arrangement of its constituents. It is particular interesting to investigate ordered structures that lack periodicity, and disordered systems that still show an apparent degree of order. For instance, a Penrose tiling of the plane consist of two basic shapes, which, when arranged properly, allow for an arbitrary large tiling, but never one that exactly repeats itself. Such structures are not only fascinating from a mathematical point of view, but are physically realised in quasicrystals. These are crystals occurring in particular metal alloys which possess an intricate non-periodic order of atoms. Due to the lack of periodicity in the structure, each atom has its own individual environment, and if one looks far enough around, no two atoms will ever have exactly the same surroundings. Therefore, it is interesting to look at properties of such structures, such as the mean number of neighbours of an atom, or mean numbers of atoms at certain distances. Such quantities are related to the diffraction patters of these materials, which provide the experimental proof of the order in the atomic positions. In a more abstract setting, thinking of a structure represented by a tiling, the corresponding question is that of the mean number of vertices in the tiling that are at a certain distance from a given vertex, averaged over all possible vertices as the centres. These numbers are called the averaged shelling numbers, and they are an example of the type of properties that are investigated in this project. The calculation of these numbers turns out to be related to interesting properties of certain types of numbers, such as factorisation of numbers into prime factors, which is a topic of interest in number theory. Moreover, these numbers and similar combinatorial properties are closely related to models of interest in physics and other sciences. This makes this project interesting from a number of different perspectives, ranging from pure mathematics to applications in physics, crystallography and materials science.

### Organisations

## People |
## ORCID iD |

Uwe Grimm (Principal Investigator) |

### Publications

Baake M
(2007)

*A radial analogue of Poisson's summation formula with applications to powder diffraction and pinwheel patterns*in Journal of Geometry and Physics
Baake M
(2008)

*Similar sublattices of the root lattice A 4*in Journal of Algebra
Baake M
(2008)

*The singular continuous diffraction measure of the Thue-Morse chain*in Journal of Physics A: Mathematical and Theoretical
Baake M
(2010)

*Surprises in aperiodic diffraction*in Journal of Physics: Conference Series
Baake M
(2009)

*Kinematic diffraction is insufficient to distinguish order from disorder*in Physical Review B
Baake M
(2008)

*Coincidence rotations of the root lattice A 4*in European Journal of Combinatorics
Grimm U
(2008)

*Homometric point sets and inverse problems*in Zeitschrift für Kristallographie
Grimm U
(2008)

*On the entropy and letter frequencies of powerfree words*
Grimm U
(2008)

*On the Entropy and Letter Frequencies of Powerfree Words*in Entropy
Heuer M
(2013)

*CSLs of the root lattice $\mathbf{A_4}$*Description | The scientific outcome covers a range of topics relevant to aperiodic order, which is concerned with the study of ordered structures without periodicity. Christian Huck's work on tomography of quasicrystals is particularly noteworthy, as it extended discrete tomography from the periodic setting to the aperiodic case, and showed that uniqueness results can also be obtained in this more general setting. His results were published in six single-authored papers. As part of international collaborations with colleagues in Germany (Baake, Frettloeh, Zeiner) and Canada (Moody), Manuela Heuer and the PI advanced the knowledge on combinatorial properties of tilings and sequences, and in particular on the mathematical diffraction of such structures, which are important from the experimental perspective. Notable results include progress in the understanding of the diffraction of the pinwheel tiling, which has complete rotation symmetry and resembles that of a stochastically oriented crystalline powder, and on the homometry problem. The latter concerns structures that cannot be distinguished by kinematic diffraction, and explicit examples were constructed that show that homometric structures ranging from completely ordered to randomly disordered can share the same (continuous) diffraction. Also, a complete description of the diffraction of the Thue-Morse chain was given, which is the paradigm of a singularly continuous diffraction measure, leading to the solution of a whole class of examples. |

Exploitation Route | It is important basic science which underpins work in crystallography and materials science, in particular in the area of aperiodic crystals. |

Sectors | Other |

Description | Research activity in this project informed and inspired outreach activity such as that of grant EP/H004866/1, where resaech results were presented to the general public within the Royal Society Summer Science Exhibition 2009, and provided educational material used by the Royal Institution in their masterclass programme. |

First Year Of Impact | 2009 |

Sector | Education |

Impact Types | Cultural |