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今天的话题，不仅涉及数学、物理学、生物学、计算机科学等自然科学，也涉及哲学、美学....希望带给大家美的享受！
在此强烈推出一段TED演讲:
Margaret Wertheim: The beautiful math of coral
http://www.ted.com/talks/margaret_wertheim_crochets_the_coral_reef.html
Enjoy!
Natural beauty
Research suggests that the abstract works of artists such as Jackson Pollock are pleasing because they imitate the natural world.
Published online 13 September 2000 | Nature |
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Pollock's 'Yellow, Grey, Black'
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Jackson Pollock's paintings are aesthetically pleasing because they obey fractal rules similar to those of the natural world. So says Henrik Jeldtoft Jensen, a mathematician at London's Imperial College.
Pollock, a pioneer of 'abstract expressionism' and one of the most influential artists of the 20th century, is well known for his technique of splashing lines of paint, apparently at random, onto a canvas. Yet the end results often have a haunting, naturalistic quality to them. This is probably because these lines have fractal properties, Jensen told the British Association for the Advancement of Science's annual festival in London this week.
Jensen invokes the cauliflower to explain fractal geometry. No matter how many pieces you break a cauliflower into, all the pieces resemble a whole cauliflower. The same goes for many natural features such as forests, rivers, clouds and mountains as you home in on them.
Since Pollock's paintings have a similar fractal nature, Jensen claims that they are not truly abstract at all. Consciously or unconsciously, Pollock was imitating the patterns of nature. Jensen contrasts this with the abstract work of Kadinsky, with its harsh Euclidean geometry of triangles, lines and other shapes confined in the frame of the painting.
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Pollock's fractal style makes his paintings open and 'frame-less', Jensen argues. Like walking through a forest, the viewer expects that the scene must be continued outside the field of view. Jensen ranks Pollack's immense canvases alongside other 'open' paintings, such as Turner's 'Snowstorm'.
Scientists are also nearer to uncovering the techniques that Pollock used. Paint simply poured in a line makes a smooth streak that has none of the complex fractal edges and undulations of Pollock's work.
Pollock was known to have swung his paint back and forth like a pendulum, using a can on the end of a string with a hole punched in it. Researchers have found that if a swinging pendulum is hit with a hammer at just the right frequency (slightly less than the natural rhythm of the pendulum), its motion becomes chaotic and the paint traces out very convincing 'fake Pollocks'. The artist had no idea of fractals or chaotic motion.
Of course, none of this approaches the innate brilliance of Pollock's work. But what it does demonstrate, if nothing else, Jensen points out, is that maths is not just a boring matter of calculating numbers: it can be just as conceptual and abstract as art.
[201 WORDS]
Van Gogh painted perfect turbulence
The disturbed artist intuited the deep forms of fluid flow.
Published online 7 July 2006 | Nature | doi:10.1038/news060703-17
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Vincent van Gogh is known for his chaotic paintings and similarly tumultuous state of mind. Now a mathematical analysis of his works reveals that the stormy patterns in many of his paintings are uncannily like real turbulence, as seen in swirling water or the air from a jet engine.
Physicist Jose Luis Aragon of the National Autonomous University of Mexico in Queretaro and his co-workers have found that the Dutch artist's works have a pattern of light and dark that closely follows the deep mathematical structure of turbulent flow.
The swirling skies of The Starry Night, painted in 1889, Road with Cypress and Star (1890) and Wheat Field with Crows (1890) — one of the van Gogh's last pictures before he shot himself at the age of 37 — all contain the characteristic statistical imprint of turbulence, say the researchers.
These works were created when van Gogh was mentally unstable: the artist is known to have experienced psychotic episodes in which he had hallucinations, minor fits and lapses of consciousness, perhaps indicating epilepsy.
"We think that van Gogh had a unique ability to depict turbulence in periods of prolonged psychotic agitation," says Aragon.
In contrast, the Self-portrait with Pipe and Bandaged Ear (1888) shows no such signs of turbulence. Van Gogh said that he painted this image in a state of "absolute calm", having been prescribed the drug potassium bromide following his famous self-mutilation.
Measured chaos
Scientists have struggled for centuries to describe turbulent flow — some are said to have considered the problem harder than quantum mechanics. It is still unsolved, but one of the foundations of the modern theory of turbulence was laid by the Soviet scientist Andrei Kolmogorov in the 1940s.
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He predicted a particular mathematical relationship between the fluctuations in a flow's speed and the rate at which it dissipates energy as friction. Kolmogorov's work led to equations describing the probability of finding a particular velocity difference between any two points in the fluid. These relationships are called Kolmogorov scaling.
Aragón and colleagues looked at van Gogh's paintings to see whether they bear the fingerprint of turbulence that Kolmogorov identified. "'Turbulent' is the main adjective used to describe van Gogh's work," says Aragn. "We tried to quantify this."
Darkness and light
The researchers took digital images of the paintings and calculated the probability that two pixels a certain distance apart would have the same brightness, or luminance. "The eye is more sensitive to luminance changes than to colour changes," they say, "and most of the information in a scene is contained in its luminance."
Several of van Gogh's works show Kolmogorov scaling in their luminance probability distributions. To the eye, this pattern can be seen as eddies of different sizes, including both large swirls and tiny eddies created by the brushwork.
Van Gogh seems to be the only painter able to render turbulence with such mathematical precision. "We have examined other apparently turbulent paintings of several artists and find no evidence of Kolmogorov scaling," says Aragon.
Edvard Munch's The Scream, for example, looks to be superficially full of van Gogh-like swirls, and was painted by a similarly tumultuous artist, but the luminance probability distribution doesn't fit Kolmogorov's theory.
The distinctive styles of other artists can be described by mathematical formulae. Jackson Pollock's drip paintings, for example, bear distinct fractal patterns.
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Random Sample
Turing’s Ideas Blossom
Science 6 April 2012: Vol. 336 no. 6077 pp. 17-18
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A sunflower is more than just a pretty face: It’s a floral expression of the so-called Fibonacci sequence—1, 1, 2, 3, 5, 8, and so on, where each number is the sum of its two preceding numbers. And now, a U.K.-based project is enlisting the help of gardeners around the world to help test a theory that originated with one of history’s greatest mathematicians, Alan Turing.
In the 1950s, toward the end of his life, Turing investigated why features in nature, such as the number of spirals in which seeds grow on a sunflower, often relate to the Fibonacci sequence. Turing tried to show that simple geometry constrains new growth so that, over time, an organism’s features take on higher Fibonacci numbers. Later theoretical models supported Turing’s ideas, but they also predicted that other numbers, such as those from a more complex “Lucas” sequence, should sometimes be present instead.
“If you look at cases where sunflowers don’t have Fibonacci numbers—that’s actually where you start getting information about whether these models are true or not,” says Jonathan Swinton, a consultant systems biologist in the United Kingdom. Working with the Manchester Science Festival, Swinton is asking people to test the models by growing sunflowers and counting the number of seed spirals (http://www.manchestersciencefestival.com/connect/getinvolved/sunflowers). If these numbers don’t match the models’ predictions, scientists will know to improve the models, he says.
Project manager Erinma Ochu at the University of Manchester in the United Kingdom, says she already has participants around the world, from the United States to Palestine. “Using the public is the best way to get a big data set,” she says.
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The incomputable reality
Alan Turing put bounds on what is computable in a famous 1936 paper. The Turing machines he presented implement finite algorithms, handling data coded as real numbers. They are deterministic, but give some bizarre results. You can build a universal machine that can simulate any other Turing machine. But not every question you can ask of it has a computable answer: you cannot predict, for example, whether it ever spits out a given number or series of numbers.
By coincidence, our Newtonian view of physics faltered at about the same time as our computable view of mathematics. Lingering problems in classical physics, such as the unpredictable trajectories of three bodies following a collision, may involve incomputability. Albert Einstein’s theory of general relativity opens up a new world of computation with exotic objects such as spinning black holes. Quantum mechanics tells us that measurements are inherently uncertain.
The concept of computability is basic to modern science, from quantum gravity to artificial intelligence. It is also relevant in the everyday world, where it is useful to distinguish problems that are merely difficult to compute in practice from those that are intrinsically impossible with any machine. Incomputability should trouble economists, because breakdowns of control in chaotic markets can wreak havoc.
But disciplinary boundaries are preventing us from getting a full view of its role. Cosmetic differences may hide revealing parallels.
EMERGENT PHENOMENA
Turing was interested in the mathematics of computing and also in its embodiment — the material environment that houses it. This theme links all of his work, from machines to the brain and morphogenesis. Although many mathematicians and software engineers today see it as irrelevant, embodiment is key to explaining the physical world.
Take turbulence: a river swollen by recent rain occasionally erupts into surprising formations that we would not expect from the basic dynamics of the water flow. The reason is coherence — non-local connectivity affects the water’s motion. Turbulence, and other ‘emergent’ nonlinear phenomena, may not be computable with a Turing machine. Zebra stripes and tropical-fish patterns, which Turing described in 1952–54 with his differential equations for morphogenesis, arise similarly.
Even in nonlinear systems, such high-order behaviour is causal — one phenomenon triggers another. Levels of explanation, from the quantum to the macroscopic, can be applied. But modelling the evolution of the higher-order effects is difficult in anything other than a broad-brush way. Such problems infiltrate all our models of the natural world.
The Universe is like that turbulent stream — its behaviour as a whole guided by myriad connections at various scales. It has many emergent levels of causality, bridged by phase transitions. The mechanistic structure that science deals with so well, and its invariant laws, are hard to explain in terms of the quantum level. Biology emerges from the quantum world, but is not computable from it. We are part of an organic whole — fragmented but coherent.
Across these boundaries, higher-level relations can feed back into lower ones. But looking up from a lower level, the causality will not be computable. For example, the uncertainty principle prevents the quantum world from fully describing the state of a particle at any instant. A measurement produces a full description, but we cannot compute how it does it. In Turing’s world, a description of reality is not always enough for a computable prediction.
Nature presents us with new ways of computing, from the Universe to the brain. Turing went on to build logical hierarchies to better understand real-world computation, which includes intuitive or unpredictable leaps. Researchers experimenting with intelligent machines today see the possibilities in such an approach. But problems of control of higher-order behaviours still present formidable challenges to implementing it.
BRIDGE BUILDING
It took nature millions of years to build a human brain. Meanwhile, we have to live with the stupidity of purely algorithmic processes. We need to embrace more experimental approaches to computation, and a renewed respect for embodied computing — as anticipated in Turing’s late work in the 1950s on artificial intelligence and morphogenesis.
Bridges between mathematicians and physicists are important if we are to do this. It is a long time since Kurt Gödel and Albert Einstein chatted in the halls of Princeton University in New Jersey. Mathematicians can bring to the table Turing’s model of basic causal structure. This would help physicists to discover more complete descriptions of the Universe — making redundant Hugh Everett’s many-worlds interpretation and related multiverse hypotheses — and fix the arbitrariness of parts of the standard model of particle physics.
Samson Abramsky, a computer scientist at the University of Oxford, UK, recently asked: “Why do we compute?” Turing computation does not create anything that is not there already in the initial data. Can information increase in computation?
If we look at the world with new eyes, allowing computation full expression, we may come to startling conclusions. ■
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Original article from: 23 FEBRUARY 2012 | VOL 482 | NATURE | 465
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发表于 2012-4-9 09:54:23
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发表于 2012-4-9 10:23:17
