A Beautiful Mind

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Topology behind Nobel Prize in Chemistry too!

From http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2016/popular-chemistryprize2016.pdf

….
With the help of this revolutionary method, Sauvage reinvigorated the field of topological chemistry, in which researchers – often using metal ions – interlock molecules in increasingly complex structures, from long chains to complicated knots. Jean-Pierre Sauvage and J. Fraser Stoddart (we will return to him soon) are leaders in this field and their research groups have created molecular versions of cultural symbols such as the trefoil knot, Solomon’s knot and the Borromean rings.

See the whole article at http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2016/popular-chemistryprize2016.pdf

Topology behind the Nobel Prize in Physics 2016

Article from http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html

Nobel Prize in Physics 2016

4 October 2016

The Royal Swedish Academy of Sciences has decided to award the Nobel Prize in Physics 2016 with one half to

David J. Thouless
University of Washington, Seattle, WA, USA

and the other half to

F. Duncan M. Haldane
Princeton University, NJ, USA

and

J. Michael Kosterlitz
Brown University, Providence, RI, USA

”for theoretical discoveries of topological phase transitions and topological phases of matter”

They revealed the secrets of exotic matter

This year’s Laureates opened the door on an unknown world where matter can assume strange states. They have used advanced mathematical methods to study unusual phases, or states, of matter, such as superconductors, superfluids or thin magnetic films. Thanks to their pioneering work, the hunt is now on for new and exotic phases of matter. Many people are hopeful of future applications in both materials science and electronics.

The three Laureates’ use of topological concepts in physics was decisive for their discoveries. Topology is a branch of mathematics that describes properties that only change step-wise. Using topology as a tool, they were able to astound the experts. In the early 1970s, Michael Kosterlitz and David Thouless overturned the then current theory that superconductivity or suprafluidity could not occur in thin layers. They demonstrated that superconductivity could occur at low temperatures and also explained the mechanism, phase transition, that makes superconductivity disappear at higher temperatures.

In the 1980s, Thouless was able to explain a previous experiment with very thin electrically conducting layers in which conductance was precisely measured as integer steps. He showed that these integers were topological in their nature. At around the same time, Duncan Haldane discovered how topological concepts can be used to understand the properties of chains of small magnets found in some materials.

We now know of many topological phases, not only in thin layers and threads, but also in ordinary three-dimensional materials. Over the last decade, this area has boosted frontline research in condensed matter physics, not least because of the hope that topological materials could be used in new generations of electronics and superconductors, or in future quantum computers. Current research is revealing the secrets of matter in the exotic worlds discovered by this year’s Nobel Laureates.

Read more at http://www.nobelprize.org/nobel_prizes/physics/laureates/2016/press.html

Geometrical Forms in Gaudi’s work

Taken from https://en.wikipedia.org/wiki/Antoni_Gaud%C3%AD#Geometrical_forms

Antoni Gaudi was as Spanish Catalan architect.

Geometrical forms

The nave in the Sagrada Familia with a hyperboloid vault. Inspiration from nature is taken from a tree, as the pillar and branches symbolise trees rising up to the roof.

This study of nature translated into his use of ruled geometrical forms such as the hyperbolic paraboloid, the hyperboloid, the helicoid and the cone, which reflect the forms Gaudí found in nature.[59] Ruled surfaces are forms generated by a straight line known as the generatrix, as it moves over one or several lines known as directrices. Gaudí found abundant examples of them in nature, for instance in rushes, reeds and bones; he used to say that there is no better structure than the trunk of a tree or a human skeleton. These forms are at the same time functional and aesthetic, and Gaudí discovered how to adapt the language of nature to the structural forms of architecture. He used to equate the helicoid form to movement and the hyperboloid to light. Concerning ruled surfaces, he said:

Paraboloids, hyperboloids and helicoids, constantly varying the incidence of the light, are rich in matrices themselves, which make ornamentation and even modelling unnecessary.[60]

Another element widely used by Gaudí was the catenary arch. He had studied geometry thoroughly when he was young, studying numerous articles about engineering, a field that praised the virtues of the catenary curve as a mechanical element, one which at that time, however, was used only in the construction of suspension bridges. Gaudí was the first to use this element in common architecture. Catenary arches in works like the Casa Milà, the Teresian College, the crypt of the Colònia Güell and the Sagrada Família allowed Gaudí to add an element of great strength to his structures, given that the catenary distributes the weight it regularly carries evenly, being affected only by self-canceling tangential forces.[61]

Gaudí evolved from plane to spatial geometry, to ruled geometry. These constructional forms are highly suited to the use of cheap materials such as brick. Gaudí frequently used brick laid with mortar in successive layers, as in the traditional Catalan vault, using the brick laid flat instead of on its side.[62] This quest for new structural solutions culminated between 1910 and 1920, when he exploited his research and experience in his masterpiece, the Sagrada Família. Gaudí conceived the interior of the church as if it were a forest, with a set of tree-like columns divided into various branches to support a structure of intertwined hyperboloid vaults. He inclined the columns so they could better resist the perpendicular pressure on their section. He also gave them a double-turn helicoidal shape (right turn and left turn), as in the branches and trunks of trees. This created a structure that is now known as fractal.[63] Together with a modulation of the space that divides it into small, independent and self-supporting modules, it creates a structure that perfectly supports the mechanical traction forces without need for buttresses, as required by the neo-Gothic style.[64] Gaudí thus achieved a rational, structured and perfectly logical solution, creating at the same time a new architectural style that was original, simple, practical and aesthetic.