In the Imaging maths series Plus explored how to visualise, transform and unfold strange geometric objects like the Klein bottle or the Möbius strip. Now Plus has come across a British artist who does just that, but without the aid of computers. John Pickering bases his work on one simple mathematical
transformation, the inversion in a circle. His works are beautifully intricate 3D objects made up of 2D slices, with each slice and its relationship to the others worked out using honest and rigorous co-ordinate geometry.
Unfortunately, Pickering's work can't be seen on the web, so if you're interested in mathematical art, keep an eye out for one of his exhibitions. Alternatively, there is a book on Pickering's art called Mathematical form: John Pickering and the architecture of the inversion principle, published by the Architectural Association.
Leonhard Euler, born on the 15th of April 1707 in Basel, Switzerland, would have turned 300 yesterday! In the current issue Plus kicks of a celebratory series of articles with Robin Wilson's "Read Euler, read Euler, he is the master of us all.". Here is a taster of Wilson's article.
Euler was the most prolific mathematician of all time. He wrote more than 500 books and papers during his lifetime — about 800 pages per year — with an incredible 400 further publications appearing posthumously. His collected works and correspondence are still not completely published: they already fill over seventy large volumes, comprising tens of thousands of pages.
Euler worked in an astonishing variety of areas, ranging from the very pure — the theory of numbers, the geometry of a circle and musical harmony — via such areas as infinite series, logarithms, the calculus and mechanics, to the practical — optics, astronomy, the motion of the Moon, the sailing of ships, and much else besides. Indeed, Euler originated so many ideas that his successors have
been kept busy trying to follow them up ever since. Not surprisingly, many concepts are named after him: Euler's constant, Euler's polyhedron formula, the Euler line of a triangle, Euler's equations of motion, Eulerian graphs, Euler's pentagonal formula for partitions, and many others.
Euler's career took him from his native Basel to the Academy at St Petersburg, where he eventually became Professor of Mathematics, to the Berlin Academy, on invitation from Prussia's Frederick the Great, and finally back to St Petersburg. He fathered thirteen children, of whom only five survived to adolescence, and reportedly carried out mathematical researches with a baby on his lap.
He died in St Petersburg on the 18th of September 1783. In a eulogy by the Marquis de Condorcet, we read about his final afternoon:
On the 7th of September 1783, after amusing himself with calculating on a slate the laws of the ascending motion of air balloons, the recent discovery of which was then making a noise all over Europe, he dined with Mr Lexell and his family, talked of Herschel's planet (Uranus), and of the calculations which determine its orbit. A little after, he called his grandchild, and fell a playing
with him as he drank tea, when suddenly the pipe, which he held in his hand, dropped from it, and he ceased to calculate and to breathe.
The distinguished mathematician Paul Cohen sadly died on Friday the 23rd of March 2007, just a few days before his 73rd birthday.
Cohen worked on a range of topics, but is best-known for his work on set theory. His work in this area cuts right down to the foundations of mathematics. The mathematician David Hilbert famously believed that it should be possible to phrase all of mathematics in a single and completely formal theory.
Based on a collection of axioms — pre-determined facts that are so self-evident they do not themselves need to be proved — and the rules of logic, it should be possible to formally prove every mathematical truth and to arrive at a complete theory that is free from contradiction.
Set theory provides a language in which such a formal system might be phrased. In maths a set is simply a collection of objects. The objects themselves are allowed to remain abstract and so you can talk about all mathematical objects — whether they are functions, numbers or anything else — in terms of sets. In the beginning of the twentieth century, mathematicians laid down a list of axioms
and rules of logic that resulted in a formal and rigorous theory of sets. The system is known as ZF theory, after Ernst Zermelo and Abraham Fraenkel.
On the face of it, sets are simple objects. But once you allow them to have an infinite number of elements, things become complicated. If you take the set of whole numbers, for example, and compare it to the set of all numbers, you'll notice that although both sets are infinite, they are fundamentally different from each other. The set of whole numbers consists of isolated objects,
whereas you can think of all the numbers as merging together to give a continuum. In some sense, the set of all numbers is "bigger" than the set of whole numbers, so we need a notion of size for infinite sets also. The mathematician Georg Cantor (1845 - 1918) formalised such a notion, called cardinality, and
in doing so came up with a conjecture that became known as the continuum hypothesis: that there is no set that is "larger" than the set of whole numbers and "smaller" than the set of all numbers. However, a proof of this fact illuded Cantor and the continuum hypothesis became the first on Hilbert's list of mathematical challenges for the 20th century.
When it comes to the axioms of set theory, things aren't all that clear-cut either. Some axioms are clear as day, for example the one which states that two sets are the same if all their elements are the same. Others are more controversial though, and one of them is the axiom of choice. It states that if you have a collection of sets, then you can form a new set by picking one element
from each. Again, this is clearly possible when you've got a finite collection of sets, but if there are infinitely many, it's not clear that a mechanism for picking an element from each always exists. Although the majority of mathematicians accept the axiom of choice, there is a school of thought which doubts it.
Paul Cohen proved two significant — and to the Hilbert school of thought disappointing — results in this area. He showed that neither the continuum hypothesis nor the axiom of choice can be proved from the axioms of ZF theory. Together with results previously proved by Kurt Gödel, this means that neither can be proved to be either true or false within ZF theory. Within ZF theory, the continuum
hypothesis will forever remain a mystery. As far as the axiom of choice is concerned, the result means that there's no clear indication as to whether you should include it or its negation as one of your initial axioms of set theory. In both cases you come up with a sound system, so the decision whether to include it or not has to be made on different grounds.
In proving these results, Cohen not only contributed to the philosophical debate about the foundations of maths, but also developed a whole new set of tools to deal with questions on what can and cannot be proved within formal mathematical systems. His work was honoured with a Fields medal in 1966. The world of maths has lost one of its most distinguished members.
We have recently come across a great range of career magazines for students and graduates from different backgrounds. KAL (for ethnic minority university students and graduates), Number Ten (for female university students and graduates), The Arberry Profile (for disabled
university students and graduates), Spectrum (for ethnic minority secondary school students) and Mint (for female secondary school students) are produced by the people at Arberry Pink.
Each magazine contains information about different areas of employment, with interviews with people working in these fields and information on employers. The latest issue of Spectrum has a feature on science and engineering, which talks to people from the British Antarctic Survey, a space mission scientist, and a geochemist.
Also the magazines are all written and edited by a team of students, in collaboration with the staff at Arberry Pink. So if you are looking for experience in journalism why not consider getting involved?
As part of the Cambridge Science Festival, the Centre for Mathematical Sciences (home of Plus) is having an open day on Saturday 24 March.
You can come and meet some of the Cabridge mathematicians who work on everything from gravity and black holes to climate change, disease dynamics and how bacteria swim. There will be hands-on activities, demonstrations, computer models and displays share some of the wonders of mathematics and theoretical physics. And the Plus team will be there so come and say hello to us at the MMP
When: Saturday 24th March 2007, 12.00 - 4.00 pm
Where: Centre for Mathematical Sciences, Clarkson Road, Cambridge
No booking required - drop in throughout the day. Limited on-site parking is available - the CMS car park is on Wilberforce Road, off Madingley Road. The Maths Cafe will be open from 12 noon until 3.30pm serving tea, coffee, soft drinks, sandwiches and snacks.
And while you there, why not catch one of these talks. No ticket is necessary, just turn up in good time to secure a seat.
CURVED SURFACES - Popular lecture by Prof Alan Beardon.
Every point of an orange can be in contact with a table top; why is the same not true of a banana? Why is it more difficult to wrap a football in paper than it is to wrap a box in paper? How do we represent the curved surface of the earth on a flat piece of paper? How do we navigate around the surface of the earth?
Saturday 24th March 2007, 12.15 - 1.15 pm
AVALANCHE! - Popular lecture by Dr Jim McElwaine
More than a million avalanches happen throughout the world every year. Most fall harmlessly, but the largest can destroy whole towns and kill thousands. This non-technical multimedia talk describes one mathematician's efforts to understand snow avalanches, from investigating disasters in the Japanese mountains to dropping half a million ping-pong balls down a ski jump.
Figures recently released by the Universities and Colleges Admissions Service (UCAS) show that 10 per cent more applications were made this year for courses in the mathematical sciences than last year. This pushed mathematics in to the top 20 most popular courses at UK universities.
Sir David Wallace, chair of the Council for Mathematical Sciences, said, "This is not just great news for the mathematical sciences. It is also great news for the UK as a whole as we need mathematics graduates more than ever before. As a bedrock for our knowledge economy, mathematics is vital for scientific research and development, and for our economic and
This year, by the closing date for applications to study in 2007, UCAS had received 33,790 applications for mathematics courses. This demonstrates a 10 per cent increase on this time last year and a massive 37 per cent increase since 2004.
Applications to the mathematical sciences dropped suddenly in 2000 following the introduction of AS-levels. The numbers have been steadily increasing ever since but are still not at the levels they were just ten years ago.