Lyapunov Exponents and Spectral Properties of Aperiodic Structures
Lead Research Organisation:
The Open University
Department Name: Faculty of Sci, Tech, Eng & Maths (STEM)
Abstract
Order and disorder are familiar concepts to humans. Our brains have evolved to recognise and to appreciate aspects of symmetry and order in nature, as well as in architecture, arts or music. Essentially, science is about detecting and describing ordered patterns in the world around us, and understanding them in mathematical terms.
It is thus surprising that it appears to be difficult to give a precise mathematical definition of the concept of order, and indeed there is currently no generally accepted definition available. Consequently, we lack a complete understanding of what types of order are possible, and how to classify them.
It turns out to be useful to take a guide from nature. The type of order considered in this project is inspired by physics and crystallography, more precisely the surprising existence of intricately ordered materials called quasicrystals. Their discovery was acknowledged with the award of a Nobel Prize in Chemistry in 2011. As an abstraction of the order of atoms in such materials, mathematicians have investigated the order in patterns or tilings of space.
The project will consider a particular type of tilings, which are based on specific rules, in which a tiling can be constructed recursively. Some of these rules lead to particularly nice tilings, which are reasonably well understood. Here we mainly concentrate on tilings that lie outside this class, and investigate their properties, using a novel approach. This promises to provide new insight into order properties of such tilings, which would be a big step towards a classification of order in spatial structures, and will shed new light on the so-called Pisot substitution conjecture, one of the long-standing conjectures in the field that has so far eluded a proof.
The new approach also provides a link between two very different characterisations of aperiodic tilings by means of spectral properties. One of these is linked to diffraction as a measure of order, which uses a concept from crystallography and is intimately linked to a mathematical concept of spectrum used in dynamical systems theory. The other spectral characterisation is inspired by studying the physics of electron transport in aperiodic structures, and considers the electronic energy spectrum. These two spectral quantities behave rather differently, and the aim of the project is to understand this and relate these to each other.
While the proposed research is fundamental in nature, order phenomena are ubiquitous in nature and an improved understanding of order will be useful in many areas of science. Also, there are many potential applications of aperiodic structures of this type. The most promising are probably in manufactured materials, such as new light-weight strong materials with designed properties that could be used in engineering or medical applications. Aperiodic tilings often are also aesthetically appealing and have increasingly been used in arts and in architecture.
It is thus surprising that it appears to be difficult to give a precise mathematical definition of the concept of order, and indeed there is currently no generally accepted definition available. Consequently, we lack a complete understanding of what types of order are possible, and how to classify them.
It turns out to be useful to take a guide from nature. The type of order considered in this project is inspired by physics and crystallography, more precisely the surprising existence of intricately ordered materials called quasicrystals. Their discovery was acknowledged with the award of a Nobel Prize in Chemistry in 2011. As an abstraction of the order of atoms in such materials, mathematicians have investigated the order in patterns or tilings of space.
The project will consider a particular type of tilings, which are based on specific rules, in which a tiling can be constructed recursively. Some of these rules lead to particularly nice tilings, which are reasonably well understood. Here we mainly concentrate on tilings that lie outside this class, and investigate their properties, using a novel approach. This promises to provide new insight into order properties of such tilings, which would be a big step towards a classification of order in spatial structures, and will shed new light on the so-called Pisot substitution conjecture, one of the long-standing conjectures in the field that has so far eluded a proof.
The new approach also provides a link between two very different characterisations of aperiodic tilings by means of spectral properties. One of these is linked to diffraction as a measure of order, which uses a concept from crystallography and is intimately linked to a mathematical concept of spectrum used in dynamical systems theory. The other spectral characterisation is inspired by studying the physics of electron transport in aperiodic structures, and considers the electronic energy spectrum. These two spectral quantities behave rather differently, and the aim of the project is to understand this and relate these to each other.
While the proposed research is fundamental in nature, order phenomena are ubiquitous in nature and an improved understanding of order will be useful in many areas of science. Also, there are many potential applications of aperiodic structures of this type. The most promising are probably in manufactured materials, such as new light-weight strong materials with designed properties that could be used in engineering or medical applications. Aperiodic tilings often are also aesthetically appealing and have increasingly been used in arts and in architecture.
Planned Impact
The proposed research is of a fundamental nature, so any application has to be seen as a long-term perspective. As with any fundamental research, it is of prime importance to inform the community working on potential applications about any developments and progress. This will be done by publishing in appropriate journals, and by attending the interdisciplinary conferences where experts meet. Publications will be made available to the wider public prior to publication via the arXiv, and subsequently by using open access journals and via the Open University research repository (ORO).
There are three main areas of (established or potential) impact of research in this field, to which the proposed project could contribute.
The first area of applications concerns technological applications of aperiodic systems. Purpose-made materials with designed properties have wide potential applications. These are not limited to naturally occurring structures, and with the possibility to manipulate systems at an atomic scale, there is a plethora of possible applications. Aperiodic structures have interesting properties and are likely to be superior to periodic and random structures in some situations (for instance due to their higher symmetry). The PI is currently involved in a project with colleagues in Design to investigate this in the context of additive manufacture of cellular structures. For aperiodic structures at the nanoscale, diffraction techniques will become relevant, and the results of this project may become important. Also, there are a number of potential applications around electronic spectral properties, such as for optical and electronic filters.
A second area of potential impact is in inspiring creative design. Some concrete examples are detailed in the Pathways to Impact section. This may be less likely to arise from this particular project, since the emphasis is on spectral properties not new structures, but it is conceivable that some particularly appealing structures may arise in connection with the final theme of this project, which could be of interest. The PI hopes to establish further contacts to artists through including arts content in the forthcoming ICMS workshop.
Public engagement with mathematics is important, in particular with regard to enthusing young people to take up mathematics at school and higher education level. There is a considerable shortage of mathematicians in the UK, particularly (but not only) in education, and the only way to change this is a long-term strategy to increase the take-up of the subject. Aperiodic order is an ideal field for engagement activities, due to its aesthetic appeal and the possibility to explain complex mathematical theory in terms of pictures in a way that makes the ideas accessible to a lay audience. This impact is very likely to be achieved, as the PI has a strong track record in public engagement, and will continue to take an active approach towards engagement. The Open University and the School of Mathematics and Statistics are very supportive of outreach activities, and the School is currently further developing its proactive approach in this area. Concerning the proposed research, the aspect that is most likely to be amenable for public engagement is that of characterising different types of (aperiodic) order, and distinguishing them by properties that are linked to applications.
There are three main areas of (established or potential) impact of research in this field, to which the proposed project could contribute.
The first area of applications concerns technological applications of aperiodic systems. Purpose-made materials with designed properties have wide potential applications. These are not limited to naturally occurring structures, and with the possibility to manipulate systems at an atomic scale, there is a plethora of possible applications. Aperiodic structures have interesting properties and are likely to be superior to periodic and random structures in some situations (for instance due to their higher symmetry). The PI is currently involved in a project with colleagues in Design to investigate this in the context of additive manufacture of cellular structures. For aperiodic structures at the nanoscale, diffraction techniques will become relevant, and the results of this project may become important. Also, there are a number of potential applications around electronic spectral properties, such as for optical and electronic filters.
A second area of potential impact is in inspiring creative design. Some concrete examples are detailed in the Pathways to Impact section. This may be less likely to arise from this particular project, since the emphasis is on spectral properties not new structures, but it is conceivable that some particularly appealing structures may arise in connection with the final theme of this project, which could be of interest. The PI hopes to establish further contacts to artists through including arts content in the forthcoming ICMS workshop.
Public engagement with mathematics is important, in particular with regard to enthusing young people to take up mathematics at school and higher education level. There is a considerable shortage of mathematicians in the UK, particularly (but not only) in education, and the only way to change this is a long-term strategy to increase the take-up of the subject. Aperiodic order is an ideal field for engagement activities, due to its aesthetic appeal and the possibility to explain complex mathematical theory in terms of pictures in a way that makes the ideas accessible to a lay audience. This impact is very likely to be achieved, as the PI has a strong track record in public engagement, and will continue to take an active approach towards engagement. The Open University and the School of Mathematics and Statistics are very supportive of outreach activities, and the School is currently further developing its proactive approach in this area. Concerning the proposed research, the aspect that is most likely to be amenable for public engagement is that of characterising different types of (aperiodic) order, and distinguishing them by properties that are linked to applications.
Organisations
Publications
Nagai Y
(2021)
Absence of absolutely continuous diffraction spectrum for certain S-adic tilings
in Nonlinearity
Bustos Á
(2022)
Admissible Reversing and Extended Symmetries for Bijective Substitutions
in Discrete & Computational Geometry
Berthé V
(2022)
Coboundaries and eigenvalues of finitary S-adic systems
Baake M
(2020)
Diffraction of a model set with complex windows
in Journal of Physics: Conference Series
Baake M.
(2020)
Fourier Transform of Rauzy Fractals and Point Spectrum of 1D Pisot Inflation Tilings
in Documenta Mathematica
Baake Michael
(2020)
FOURIER TRANSFORM OF RAUZY FRACTALS AND POINT SPECTRUM OF 1D PISOT INFLATION TILINGS
in DOCUMENTA MATHEMATICA
Grimm U
(2021)
Gaussian orthogonal ensemble for quasiperiodic tilings without unfolding: r -value statistics
in Physical Review B
Grimm U
(2021)
Highly symmetric aperiodic structures -INVITED
in EPJ Web of Conferences
Description | As a first outcome of the funded research, we gained a better understanding of the behaviour of the diffraction of aperiodically ordered structures near the origin, which has been argued to constitute a measure of order in the system, associated with the term "hyperuniformity". There has also been progress on the understanding of spectral properties of inflation-based structures, exploiting their built-in self-similarity by a renormalisation approach. Several joint papers with colleagues in Germany and the US explored a matrix cocycle approach to prove the absence of absolutely continuous spectral components for large classes of inflation structures. Generalisations of these results to S-adic systems have been completed with the PDRA on this project. Further investigations are under way for spectra of S-adic systems. A new key observation is that for systems which possess an inflation as well as a cut and project description the cocycle approach can be mirrored in internal space, providing new insight in particular for cases where the windows of the projection are Rauzy fractals. We have also investigated the discrete spectrum of a new family of S-adic systems, which we call torsion-free S-adic systems. |
Exploitation Route | There is a lot of activity around hyperuniformity, and this work puts some of the heuristic results on a firm footing. Results on spectral properties are relevant for applications in nanooptics and other areas. |
Sectors | Other |
Description | ICMS Public engagement activity. (Title: An art exhibit in honour of Uwe Grimm) |
Amount | £9,800 (GBP) |
Organisation | International Centre for Mathematical Sciences |
Sector | Public |
Country | United Kingdom |
Start | 05/2022 |
End | 07/2022 |
Description | LMS Conference grant (Scheme 1) |
Amount | £5,500 (GBP) |
Funding ID | 12120 |
Organisation | London Mathematical Society |
Sector | Academic/University |
Country | United Kingdom |
Start | 05/2022 |
End | 07/2022 |
Description | Novel superior materials based on aperiodic tilings |
Amount | £201,913 (GBP) |
Funding ID | EP/V047108/1 |
Organisation | Engineering and Physical Sciences Research Council (EPSRC) |
Sector | Public |
Country | United Kingdom |
Start | 04/2021 |
End | 01/2023 |
Description | A meeting in honour of Uwe Grimm |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Study participants or study members |
Results and Impact | This event, which commemorated Uwe Grimm's research consisted of a research meeting, bringing together 50 researchers in aperiodic tiling theory. The research meeting received funding from an LMS Scheme 1 conference grant, Bielefeld University and the Open University. |
Year(s) Of Engagement Activity | 2022 |
URL | https://www.open.ac.uk/stem/mathematics-and-statistics/aperiodic-tilings-2022#research-meeting |
Description | Aperiodic Tilings at Maths Fest (Kensington) |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Schools |
Results and Impact | Activities on Aperiodic Tilings forming part of the Open University stall in the "Maths Village" at Maths Fest 2020, which is "is an all-day extravaganza for sixth-form students (and maths-keen KS4) to celebrate and be inspired by mathematics". |
Year(s) Of Engagement Activity | 2020 |
URL | http://maths-fest.com/ |
Description | Aperiodic tilings, An exhibition in honour of Uwe Grimm |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Schools |
Results and Impact | This was a mathematical art exhibition inspired by aperiodic tilings, in honour of Uwe Grimm. This event, which commemorated Uwe Grimm's passion for outreach. It consisted of an art exhibition and an interactive workshop on symmetries and tessellations, attracting 750 visitors, of which 450 were school students in Key stage three (years 7-9). The art exhibition was funded by an ICMS Public Engagement Activity award. The event website https://www.open.ac.uk/stem/mathematics-and-statistics/aperiodic-tilings-gallery includes an online video of artwork from the exhibit. |
Year(s) Of Engagement Activity | 2020,2022 |
URL | https://www.open.ac.uk/stem/mathematics-and-statistics/aperiodic-tilings-gallery |
Description | Exhibition on Tilings and Einstein's hat, British Science Festival, Exeter, September 2023 |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | National |
Primary Audience | Public/other audiences |
Results and Impact | Quoting from its website, "The British Science Festival is is all about making scientific research and innovation relevant and accessible to everyone...At the same time we believe that involving more people in science improves outcomes, more viewpoints, creativity, ideas and contexts make to for better results." On Friday 8th September, the Open University and Queen Mary University of London had a "stand" in the "Forum Street" with a display of periodic and aperiodic tilings. The intend purpose was to educate the general public on aperiodic tilings, recent advances with the discovery of the monotile. There were about 100-150 visitors with which I had a conversation about tilings. |
Year(s) Of Engagement Activity | 2023 |
URL | https://britishsciencefestival.org/events |
Description | Kieler Woche der Mathematik, Kiel, Germany, 11 October 2019 |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | International |
Primary Audience | Schools |
Results and Impact | Invited presentation at the end of the "Kieler Woche der Mathematik", an annual week of program at the University of Kiel in Germany, aimed at school pupils from Year 9 onwards. The 2019 topic was "Parkettierungen der Ebene" which translates to "Planar Tilings". |
Year(s) Of Engagement Activity | 2019 |
URL | https://www.uni-kiel.de/de/detailansicht/news/kw-mathematik/ |
Description | One-day Stand, "Monotiles and Einstein's hat" Festival of communities, QMUL |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | Local |
Primary Audience | Public/other audiences |
Results and Impact | From the festival's website: "The Festival of Communities returned on 10th and 11th June for a weekend of activities, games and glorious sunshine. Across both days over 7,000 visitors joined Queen Mary researchers and local community groups to celebrate everything that makes Tower Hamlets a fantastic place to live and work.". We participated in this festival for the day of the 11th of June. The intended purpose of this festival is to engage with the local community, exposing it to the kind of research we do at QMUL, and fostering interest in academic matters in children. Hundreds of families visited our stand, and engaged with "jigsaw puzzles" consisting of aperiodic tiles. There were several engaged and interesting discussions with the community, who were able to see a hands on expression of modern mathematical research and recent advances. |
Year(s) Of Engagement Activity | 2023 |
URL | https://www.qmul.ac.uk/festival/about/2023/ |
Description | Open University Masterclass |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Schools |
Results and Impact | This was part of the Open University Mathematics Masterclass series. Along with Charlotte Webb, I gave an online 90 minute masterclass to schoolchildren aged 11-15, on tessellations, tilings, and quasi crystals. We gave a background on the area, and interspersed it with some breakout sessions consisting of puzzles/tessellation design. The students asked and also answered questions in the chat, and there was a large discussion about quasicrystals. |
Year(s) Of Engagement Activity | 2022 |
URL | http://mcs.open.ac.uk/RI_MasterClasses/ |
Description | Participation at Maths Fest 2022, Royal Institution |
Form Of Engagement Activity | Participation in an open day or visit at my research institution |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Schools |
Results and Impact | I presented examples of aperiodic tilings to engage the A-level students and explain to them a little about the theory of aperiodic tilings. |
Year(s) Of Engagement Activity | 2022 |
URL | https://amsp.org.uk/events/details/9226 |
Description | Research in Prison: Presentation at HMP Stafford |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Local |
Primary Audience | Public/other audiences |
Results and Impact | Presentation at HMP Stafford within the OUResearch project, which has been running since April 2018 in two prisons in The West Midlands - HMP Stafford & HMP Oakwood. Audience is prisoners engaged in educational activities. |
Year(s) Of Engagement Activity | 2019 |
Description | Royal Institution Mathematics Masterclass, Bletchley Park, 19 October 2019 |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Media (as a channel to the public) |
Results and Impact | Royal Institution Mathematics Masterclass lecture on "Patterns and Tilings". One lecture in a series of six lectures aiming to stimulate and encourage young people in the art and practice of mathematics and to develop a sense of enjoyment in the subject. Audience consisted of about 35 pupils (Year 9) from regional schools, as well as a teacher and a STEM ambassador. |
Year(s) Of Engagement Activity | 2019 |
URL | http://mcs.open.ac.uk/RI_MasterClasses/bletchley.php |
Description | Talk on maths research, Girls in Maths taster Day, QMUL |
Form Of Engagement Activity | Participation in an activity, workshop or similar |
Part Of Official Scheme? | No |
Geographic Reach | Local |
Primary Audience | Schools |
Results and Impact | The purpose of my talk was to give a female A-level student audience a sense of what mathematical research involves. The intended purpose was to foster interest in further study of mathematics, and to encourage females think of mathematics research as an option. |
Year(s) Of Engagement Activity | 2024 |
URL | https://www.qmul.ac.uk/maths/news-and-events/events-/items/girls-in-maths-taster-day-.html |
Description | masterclass, online, entitled "Tilings and Patterns", destined for year 10 students |
Form Of Engagement Activity | A talk or presentation |
Part Of Official Scheme? | No |
Geographic Reach | Regional |
Primary Audience | Schools |
Results and Impact | The purpose of this 90 minute masterclass was to introduce students to regular and aperiodic tilings in nature and mathematics. With elementary notions, we gave the students a flavour of the research area, and how it is linked to numeration systems. We answered questions and completed two exercises in breakout rooms. |
Year(s) Of Engagement Activity | 2020,2022 |
URL | http://mcs.open.ac.uk/RI_MasterClasses/bletchley.php |