Escher’s “Circle Limit III”
At our third “Maths Beyond the Syllabus” session, on Wednesday 15 November, Sarah Hart (professor of maths at Birkbeck, University of London) gave us a glance into where geometry meets art.
We welcomed students from Newham Collegiate Sixth Form, Wimbledon High School, and Sydenham School to join us.
Maurits Escher was a Dutch artist who, after visiting and marvelling at the Islamic art of the tilings at the Alhambra in Granada, Spain, in 1937 started producing a new sort of artwork making heavy use of the mathematical ideas of symmetry and perspective.
In 1954 the International Congress of Mathematicians was held in Amsterdam. Escher went to it and met Donald Coxeter, a mathematician of British origin who worked most of his life in Canada, and one of the leading figures in 20th century geometry.
Coxeter made Escher aware of the new sorts of symmetry he could depict using not ordinary “Euclidean” geometry, but what mathematicians call “hyperbolic geometry”. There’s a formula here: space can be tiled with regular polygons of n sides, with k of them meeting at each point:
if 1/n + 1/k equals 1/2, in an Euclidean (flat) plane;
if 1/n + 1/k > 1/2, in elliptic space (like the surface of a sphere).
if 1/n + 1/k < 1/2, in a hyperbolic plane (which is a little more difficult to visualise, but can be “squashed down” into a disk like that in Escher's artworks, a Poincaré disk).
The “Circle Limit” artwork above is an example of a regular tiling in a Poincaré disk, and Escher produced several others on the same theme. Until Escher’s death in 1972, he continued to correspond with Coxeter about geometry.
The maths here feeds not only into art, but also into many fields of physics, crystallography, chemistry, and engineering.
From the fact that geometry has disappeared from A level syllabuses, you might conclude that geometry is dead and finished. Far from it. Some areas of geometry are among the most active front lines of research today.
The Maths of Juggling; and Building a Theorem
On Wednesday 18 Oçtober we welcomed students from Newham Collegiate School, City Academy Hackney, and Enfield Grammar School to join our Further Maths students to learn about the mathematics of juggling, and Building a Theorem.
Joe Watkins from Kent University first introduced us to the “Siteswap” mathematics which have been used to define juggling patterns. Each juggling pattern is represented by a sequence of positive whole numbers indicating the heights to which balls are thrown, like 31, or 312, or 5223. Mathematical rules tell us which sequences represent a real possible juggle, and which don’t.
Though the maths was developed as recently as the 1980s, and people have been doing complicated juggling tricks for thousands of years, the maths has allowed jugglers to develop new patterns
is a video about Siteswap by Colin Wright, one of its inventors.
is another by Allen Knutson, a professor of maths at Cornell University in New York and also a former world juggling champion.
Then Dr Watkins explored the precise reasoning required to prove a typical mathematical theorem – that the only regular-polygon tiles (that is, tiles which have straight-line sides, all equal) which can be used to tile a flat surface exactly, without any gaps, are triangles, squares, and hexagons.
Topology and Big Data
On Monday 18 September we welcomed students from City of London School for Girls, City Academy Hackney, Haberdasher’s Askes, and the Charter School to join our Further Maths students to learn about Topology and Big Data.
Jacek Brodzki is professor of maths at Southampton University and leader of the Engineering and Physical Sciences Research Council project on Topology and Big Data – https://www.southampton.ac.uk/jtd/research/index.page. He gave a talk on “From Pythagoras to Big Data”, and then led the students in an interactive session in which they explored some of the basic ideas of topology.
Topology is the study of the properties of shapes which remain intact when they are stretched or squeezed or bent, but not torn. Thus the joke: a topologist is a mathematician who can’t tell the difference between a doughnut and a coffee mug.
Its use to analyse data is quite new. As Professor Brodzki pointed out, however, the use of mathematics to reason precisely and simply about complicated approximations – which is the issue here – dates back thousands of years.
Take the Pythagoras theorem: the square on the hypotenuse is equal to the sum of the squares on the other two sides. But if the two shorter sides have length 1, then the hypotenuse has length √2.
√2 cannot be, has never been, never will be, calculated exactly as a fraction or a decimal! Yet mathematics allows us to reason precisely with this number which we can never write down precisely.
Or the formula for the area of a circle: A = πr2. In ancient times Archimedes estimated π to a good enough accuracy for most modern engineering applications, by looking at a 96-sided polygon just fitting inside the circle and a 96-sided polygon just fitting outside.
For thousands of years people tried to “square the circle”, that is, to find a way of constructing a square that would have the same area as a circle. Even though √2 can not be written down exactly, at least it was possible to construct exactly a square with area 2.
In 1882 it was finally proved that it is impossible to “square the circle”. We can never construct a line of exact length √π, or π. Still, we can do many delicate and exact calculations using π.
The modern use of topology to deal with big data comes from technology which gives us unprecedentedly huge amounts of data, but complex, heterogeneous, data, of varying quality, quite different from the orderly “record the temperature every 10 seconds” type of data we may be used to in physics.
The tools of topology enable us to get precise ideas about something so apparently vague as “the shape” of the data. These tools have already produced new understanding in the study of conditions like diabetes and breast cancer.
Later that week, Derek Hill, a professor of medical imaging at UCL and Executive Director of the Ixico company, “neuroscience experts in image data management and analysis”, came to tell our Further Maths classes about what his company does.
He reported on one particular project, where they are getting more reliable estimates of how much sleep medical patients get. The hours and quality of sleep are estimated from tiny “accelerometers”, electronic gadgets available very cheaply which measure sharp movement. Ixico’s recent research has developed machine-learning methods which process the masses of data obtained from the “accelerometers” much more accurately.
For more on topology and big data, see: