Home Random Page


CATEGORIES:

BiologyChemistryConstructionCultureEcologyEconomyElectronicsFinanceGeographyHistoryInformaticsLawMathematicsMechanicsMedicineOtherPedagogyPhilosophyPhysicsPolicyPsychologySociologySportTourism






A remarkable mosaic of atoms

In quasicrystals, we find the fascinating mosaics of the Arabic world reproduced at the level of atoms: regular patterns that never repeat themselves. However, the configuration found in quasicrystals was considered impossible, and Daniel Shechtman had to fight a fierce battle against established science. The Nobel Prize in Chemistry 2011 recognizes a breakthrough that has fundamentally altered how chemists conceive of solid matter.

On the morning of April 8, 1982, an image counter to the laws of nature appeared in Daniel Shechtman's electron microscope. In all solid matter, atoms were believed to be packed inside crystals in symmetrical patterns that were repeated periodically over and over again. For scientists, this repetition was required in order to obtain a crystal.

Shechtman's image, however, showed that the atoms in his crystal were packed in a pattern that could not be repeated. Such a pattern was considered just as impossible as creating a football using only six-cornered polygons, when a sphere needs both five- and six-cornered polygons. His discovery was extremely controversial. In the course of defending his findings, he was asked to leave his research group. However, his battle eventually forced scientists to reconsider their conception of the very nature of matter.

Aperiodic mosaics, such as those found in the medieval Islamic mosaics of the Alhambra Palace in Spain and the Darb-i Imam Shrine in Iran, have helped scientists understand what quasicrystals look like at the atomic level. In those mosaics, as in quasicrystals, the patterns are regular -- they follow mathematical rules -- but they never repeat themselves.

When scientists describe Shechtman's quasicrystals, they use a concept that comes from mathematics and art: the golden ratio. This number had already caught the interest of mathematicians in Ancient Greece, as it often appeared in geometry. In quasicrystals, for instance, the ratio of various distances between atoms is related to the golden mean.

Following Shechtman's discovery, scientists have produced other kinds of quasicrystals in the lab and discovered naturally occurring quasicrystals in mineral samples from a Russian river. A Swedish company has also found quasicrystals in a certain form of steel, where the crystals reinforce the material like armor. Scientists are currently experimenting with using quasicrystals in different products such as frying pans and diesel engines.

Daniel Shechtman, Israeli citizen. Born 1941 in Tel Aviv, Israel. Ph.D. 1972 from Technion -- Israel Institute of Technology, Haifa, Israel. Distinguished Professor, The Philip Tobias Chair, Technion -- Israel Institute of Technology, Haifa, Israel.

Exotic sphere discoverer wins mathematical 'Nobel'

A sphere is a sphere, right? Yes, if you mean a globe or a beach ball – what mathematicians call a two-dimensional sphere – but not if you are talking about a sphere in seven dimensions.

Now the mathematician who discovered that spheres start to behave differently in higher dimensional space – an insight that seeded a whole new field of mathematics – has been awarded the $1 million dollar Abel prize by the Norwegian Academy of Science and Letters.



John Milnor of the Institute for Mathematical Sciences at Stony Brook University in New York, was recognised for his "pioneering discoveries in topology, geometry and algebra".

"It feels very good," Milnor told New Scientist, though he says the award was somewhat unexpected: "One is always surprised by a call at 6 o'clock in the morning."

Inflated cube

Topologists like Milnor study shapes whose mathematical properties aren't changed by stretching or twisting, but they aren't concerned with the exact geometrical properties of a particular shape, like lengths or angles. For example, you can turn a cube into a sphere by inflating it, so the two shapes are topologically identical. But you can't turn a sphere into a doughnut without tearing a hole, so they are topologically different.

It is also possible to apply stricter rules to these transformations by making them much "smoother" – what mathematicians call differentiable. For shapes in three dimensions or less, those that share a topological geometry – for example a sphere and a cube – also have the same differentiable structure.

But mathematicians also study shapes in higher dimensions – even if they're difficult to imagine. "You can often think of analogous things that are small enough to visualise," explains Milnor. "The human brain is amazingly able to tackle all sorts of things."

Tangled sphere

Milnor did just that in 1956 when he discovered a seven-dimensional mathematical object that is identical to a seven-dimensional sphere under the rules of topology, but has a different differentiable structure. He called this shape an "exotic sphere".

This was the first time a shape had been found sharing the topological properties – but not the differentiable structure – of its lower-dimensional counterpart. It led to the field that is now known as "differential topology".

What does an exotic sphere look like? It's difficult to imagine but bear in mind that it's possible to tangle up a higher-dimensional sphere in a way that isn't possible in two.

Imagine splitting an ordinary sphere into two halves along the middle, so that each half has a copy of every point on the equator. Now rejoin the two halves so that the southern copy of a point doesn't join its northern counterpoint. In two dimensions, there's only one way to do this: by twisting the sphere. But in seven dimensions the points can be mixed up with respect to each other in multiple different ways.

Smooth Poincaré

It turns out there are a total of 28 exotic spheres in seven dimensions, and they also exist in other dimensions. Dimension 15 has as many as 16,256, while others like dimensions five and six only have the ordinary sphere. Mathematicians don't yet know whether exotic spheres exist in four dimensions – a problem known as the smooth Poincaré conjecture, and related to the generalised Poincaré conjecture, which was solved in 2003.

"He's been a great inspiration to many, many mathematicians," says Timothy Gowers, a mathematician at the University of Cambridge who gave a talk on Milnor's work following the prize announcement.

Milnor is also well known for teaching other mathematicians about his idea. "Every time he writes a book, it turns into a classic," adds Gowers

 


Date: 2015-12-11; view: 830


<== previous page | next page ==>
Written in the stars | Physics Nobel goes to Serge Haroche and David Wineland
doclecture.net - lectures - 2014-2024 year. Copyright infringement or personal data (0.007 sec.)