Henri Poincaré died 100 years ago today. He is most famous for the conjecture (now theorem) which carries his name and which remained open for almost 100 years, until
Grigori Perelman announced a proof in 2003. But the conjecture isn't all there was to Poincaré. One of his teachers reportedly described him as a "monster of maths" who, perhaps because of his poor eyesight, developed immense powers of visualisation, which must
have helped him particularly in his work on geometry and topology. He has been hailed
one of the last people whose understanding of maths was truly universal. And he also thought about the philosophy of mathematics. He believed that intuition has an important
role to play in maths, and anticipated the work of Kurt Gödel, who proved
that maths cannot ever be completely formalised. Finally, and extremely pleasingly for us here at Plus, Poincaré was one of
the few scientists of his time to share his knowledge by writing numerous
popular science articles.
You can find out more about the Poincaré conjecture and related maths in these Plus articles:
So results just released from the Tevatron experiment in the US strongly support last December's announcement from the LHC – it's looking promising that the Higgs boson exists and that it has a mass around 125GeV. But as we explained in our previous news story, Countdown to the Higgs, the level of evidence produced so far doesn't count as a discovery. Physicists will only declare they have discovered the Higgs boson if they are 99.99995% confident of their result – the elusive five-sigma level.
And we might not have to wait much longer. Excitement is mounting about tomorrow's announcement of the latest results from the LHC. You can watch the seminar at 8am and press conference at 10am BST live from CERN. And to whet your appetite here are some new articles explaining exactly what the Higg's particle is, does and how they are hunting for it. We're just off to put some champagne on ice...
The notorious Higgs boson, also termed the god particle, is said to have given other particles their mass. But how did it do that? In this two-part article we explore the so-called Higgs mechanism, starting with the humble bar magnet and ending with a dramatic transformation of the early Universe.
The European Congress of Mathematics. The logo is an infinity-shaped pretzel!
Greetings from the beautiful city of Krakow, where the 6th European Congress of Mathematics opened yesterday! Around 1,000 mathematicians, donning straw hats and flip-flops to resist the incredibly hot weather, have come together here to chat, share and listen to lectures, and Plus will be reporting from the congress all week.
After that it was a conference lunch including infinity-shaped pretzels, and then the auditorium filled once more to hear Adrian Constantin's fascinating talk on water waves — it's not the water that moves with the wave, but the wave moving through the water. And when a wave breaks the maths that describes it seizes to work. I'll be talking to Constantin about his maths tomorrow and you'll be able to read an article on Plus soon. I'll also be interviewing Marta Sanz-Solé, the President of the European Mathematical Society.
Sara Santos and her maths buskers on the streets of Krakow. The challenge here was to turn the waistcoat inside out while being handcuffed.
But the congress isn't just about mathematicians talking to each other. After the lecture I went in search of Sara Santos, who has taken her mathematical buskers to the streets of Krakow, handcuffing innocent Krakowians (a punishment for dividing by zero), constructing emergency pentagons, and reading minds. But as Sara says, it's not about magic tricks but about the magical fact that the world we live in is written in the language of maths. You can hear from some maths buskers in our podcast to be published soon and if you'd like to become a maths busker yourself, visit the maths busking website.
Krakow really is as beautiful as everyone says. It's the oldest city of Poland with an amazing medieval market square, the largest in Europe. And the town has picked up the mathematical theme with no less than three art galleries showing mathematical art — if I find the time between lectures, interviews, and wine-and-canapé receptions, I'll go and visit. But now I'm off for the last task of today: investigate the specific gravity of Polish beer.
How does the uniform ball of cells that make up an embryo differentiate to create the dramatic patterns of a zebra or leopard? How come there are spotty animals with stripy tails, but no stripy animals with spotty tails? The answer comes from an ingenious mathematical model developed by Alan Turing.
Turing's scientific legacy is going stronger than ever. An example is an announcement from February this year that scientists have devised a biological computer, based on an idea first described by Turing in the 1930s.
Our planet is shaped by the oceans, the dynamic geology and the changing climate. It teems with life and we, in particular, have a massive impact as we build homes, grow food, travel and feed our ever-hungry need for energy. Mathematics is vital in understanding all of these, which is why 2013 has been declared as the year for the Mathematics of Planet Earth.
As well as encouraging research into fundamental questions about the Earth and how to meet the challenges it faces, there will also be many opportunities during 2013 for everyone to get involved including public lectures and workshops, competitions and exhibitions. The first such competition is now underway: the MPE 2013 competition to design an exhibit about the mathematics of Planet Earth.
The competition is open for submissions until 20 December 2012. The winning entries, as well as winning cash prizes, will be exhibited in institutions around the world, including the UNESCO headquarters in Paris in the inaugural exhibition in March 2013. All exhibitions will be open source and hosted by IMAGINARY, where anyone can download and reproduce the exhibits in their own museums or galleries.
But what is a cubic Hamiltonian graph, you ask? A graph, of course, is just a bunch of points (vertices) connected by lines (edges). A cubic graph is a graph where every vertex has 3 edges – that is, each vertex is connected to exactly three others in the graph. And a Hamiltonian graph is a graph which has a closed loop of edges (a cycle) that visits each vertex in the graph once and only once, (this is called a Hamiltonian cycle). So a cubic Hamiltonian graph is a graph where each vertex is joined to exactly three others and the graph has a cycle visiting each vertex exactly once.
What has made us so excited about cubic Hamiltonian graphs is watching Manning's cubic Hamiltonian graph builder in action. The builder starts from the Hamiltonian cycle in the graph. This loop of edges accounts for two out of the three edges for every vertex in the graph. The builder then starts adding in the third edge for each vertex, knitting the graph together before your eyes. The creation of a torus is particularly beautiful to watch.
Manning uses something called the LCF notation to build the graphs. This ingenious notation succinctly describes the structure of cubic Hamiltonian graphs by describing how you add the extra edge to each vertex by counting backwards or forwards around the Hamiltonian cycle. For example our torus has the LCF notation 150. This means that every vertex is joined to another 10 edges along the cycle, and this process is repeated 150 times, once for each vertex, to complete the graph. (The 1st vertex is joined to the 11th, the 2nd to the 12th, the 3rd to the 13th, and so on...)
A graph represented by the LCF notation [10,7,4,-4,-7,10,-4,7,-7,4]2 starts with a cycle of 20 vertices – inside the square brackets is a list of instructions for 10 edges, and these are repeated twice, giving the total number of extra edges, and therefore vertices, as 20. In this graph the 1st vertex is joined to the 11th (a distance of 10 edges), the 2nd to the 9th (a distance of 7), the 3rd to the 7th (a distance of 4), the 4th to the 20th (a distance of -4, or counting backwards 4 edges), the 5th to the 18th (distance of -7), and so on. This graph is knitted into a dodecahedron.
Watching Manning's program knit together these graphs is beautiful to watch. But this mathematics has many important uses as well. Not only are Hamiltonian cycles important mathematically, they also have many useful applications. You can read more about graphs on Plus, and about the role Hamiltonian cycles play in bell ringing and DNA analysis. And you can read more about Christopher Manning's work on his blog.