How do Shapes Fill Space?

Lead Research Organisation: Open University
Department Name: Mathematics & Statistics

Abstract

Shapes fill space all around us, from bathroom tilings to brick walls. The puzzling problem of how to fit shapes together so that they fill space to form what we call tilings has been considered throughout much of the history of humanity. The problem probably emerged first in the arts, where tiles were used to produce interesting patterns, like those of Islamic art. It has also been studied as part of mathematics since the ancient Greeks. For example, our understanding of symmetries and their mathematical description in terms of group theory originated from the investigation of tilings, and underlies the classification of crystal structures.The modern era for tilings began in the 1960s when Berger proved that the problem of whether a given set of shapes could tile the plane was undecidable, a result extended to the hyperbolic plane by Margenstern in 2007. This led directly to the discovery of new worlds of tiling theory and fascinating examples such as the Penrose tiling. The discovery of quasicrystals (crystals with 'forbidden' symmetry) gave additional impetus as the Penrose and related tilings provide models for non-periodic ordered structures. From a scattering of strange examples, our understanding is now evolving to coalesce into a coherent theory. In particular, the theory of substitution rules is giving a natural setting for the Penrose tiling.Tilings therefore offer a combination of deep mathematics with beautiful imagery, which makes them an ideal topic for public engagement activities. The visual appeal, the link to arts and architecture and the interactive character of related puzzle-type activities, as well as the link to current research on mysterious materials such as quasicrystals, fascinates audiences across all age groups. Because tilings are familiar objects, this topic avoids the barrier often caused by the mathematical language of symbols and equations, and enables us to communicate non-trivial mathematical concepts to a public audience.This project will create material for an exhibit at the Royal Society Summer Exhibition 2009, which is expected to attract in excess of 5000 visitors. After the exhibition in June/July 2009, the materials are adapted for continued use in UK-wide mathematics masterclasses (1 to 2.5 hour interactive sessions for 10-18 year olds) supported by the Royal Institution (Ri) and for use in Family Fun Days hosted at the Ri.

Publications

10 25 50
 
Description The Royal Society Summer Science Exhibition 2009 was very successful, attracting in excess of 5500 visitors. Our exhibit had been given an ideal position in the main hall, so all visitors will at least have passed by, and a large proportion of visitors engaged with the material, some for considerable time. Throughout the week, we had a constant stream of visitors. Despite a large team of presenters, everybody was busy explaining the exhibit essentially without pause. There was a lot of positive feedback from all age groups, and particular children spent considerable time with the hands-on puzzles and the visualisation software, while adults tended to be more interested in the current research issues, or in the connection to the arts. A typical comment of a young person, after spending 20 minutes at the exhibit, was a smiling "I'd rather do this [experimenting with the tiles] than write out feedback". Several school groups also spent extended time at the exhibit, including one who said that they'd particularly come to see our exhibit, as we were particularly interested in mathematics.
Following on from the exhibition, educational materials have been developed by Vinay Kathotia at the Ri, in close collaboration with Edmund Harris. Trial sessions of masterclasses and workshops held at schools during 2009 and 2010 reached 240 teachers and 776 students so far, and presumably more through the use of kits that have been left at schools. A couple of masterclasses, particularly aimed at sixth form students, is still being developed and tested, and will become available soon. The feedback obtained from teachers and students has been informing the development process. The "How do Shapes Fill Space?" masterclass is already one of the more popular Ri masterclasses, and is increasingly assimilated into primary and secondary classroom practice. Support by the project partners and by the Ri's other supporters has been useful in getting more materials out to schools.
Exploitation Route There is continuing use of exhibition-related materials by the Royal Institution for their masterclass programme.
Sectors Education,Other

URL http://www.tilings.org.uk/shapes/
 
Description Impact from this project was by engaging the public at the Royal Society Summer Science Exhibition, and by providing materials used by the Royal Institution for their masterclass programme.
First Year Of Impact 2009
Sector Education
Impact Types Cultural

 
Description Imperial College London 
Organisation Imperial College London
Country United Kingdom 
Sector Academic/University 
Start Year 2009
 
Description Polydron UK Ltd 
Organisation Polydron UK Ltd
Country United Kingdom 
Sector Private 
Start Year 2009
 
Description Royal Institution of Great Britain 
Organisation The Royal Institution of Great Britain
Country United Kingdom 
Sector Academic/University 
Start Year 2009
 
Description The Royal Society of London 
Organisation The Royal Society
Country United Kingdom 
Sector Academic/University 
Start Year 2009
 
Description University of Leicester 
Organisation University of Leicester
Country United Kingdom 
Sector Academic/University 
Start Year 2009
 
Description Zometool Inc 
Organisation Zometool Inc
Country United States 
Sector Private 
Start Year 2009
 
Description Royal Society Summer Science Exhibition 2009 
Form Of Engagement Activity Participation in an activity, workshop or similar
Part Of Official Scheme? Yes
Geographic Reach National
Primary Audience Public/other audiences
Results and Impact Exhibition at the Royal Society Summer Science Exhibition in London. Presenting, talking and discussing with general public, teachers and school students, as well as invited guests.

Raising awareness and interest in mathematical research, particularly among school age children. Continuing impact by website and by materials used in Ri masterclasses.
Year(s) Of Engagement Activity 2009
URL http://www.tilings.org.uk/shapes/