The shape of life

Scientists find clues to the formation of Fibonacci spirals in nature

Conical shapes with irregularities produce parastichous spiral X patterns such as on the scientists microstructure at left and a strawberry at right. Credit: Li et al. 2007 Applied Physics Letters.
Conical shapes with irregularities produce parastichous spiral ‘X’ patterns, such as on the scientists’ microstructure at left, and a strawberry at right. Credit: Li, et al. ©2007 Applied Physics Letters.

While the aesthetics and symmetry of Fibonacci spiral patterns has often attracted scientists, a mathematical or physical explanation for their common occurrence in nature is yet to be discovered. Recently, scientists have successfully produced Fibonacci spiral patterns in the lab, and found that an elastically mismatched bi-layer structure may cause stress patterns that give rise to Fibonacci spirals. The discovery may explain the widespread existence of the pattern in plants.

Chaorong Li, of the Zhejiang Sci-Tech University and the Institute of Physics in Beijing, along with Ailing Ji and Zexian Cao, both of the Chinese Academy of Sciences, produced their Fibonacci spiral pattern by manipulating the stress on inorganic microstructures made of a silver core and a silicon dioxide shell. The spontaneous assembly of Fibonacci patterns has rarely been realized in the laboratory, and the scientists’ results suggest that plant patterns might be modeled by mutually repulsive entities for both spherical and conical surfaces.

[Scientists find clues to the formation of Fibonacci spirals in nature]
“Patterns that evolve naturally are generally an optimized configuration for an assembly of elements under an interaction,” Cao explained to PhysOrg.com. “We conjecture that the Fibonacci spirals are the configuration of least elastic energy. Our experimental results provide a vivid demonstration of this energy principle. This is the best support for this energy principle of phyllotaxis (or “leaf arrangement,” often credited to D'Arcy Thompson) before a rigorous mathematical proof is available.”

Fibonacci spiral patterns appear in many plants, such as pinecones, pineapples, and sunflowers. The patterns consist of spirals that curve around a surface in both the “sinister” form (clockwise) and the “dexter” form (counterclockwise). The numbers of spirals on a surface are two consecutive numbers in the Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, etc.). For example, Li, Ji, and Cao produced a series of spirals of 3x5, 5x8, 8x13, and 13x21. Because their microstructures were very small, the next series (21x34) would have required more than 700 “spherules,” creating so much stress that the structure would break.

In their experiment, the scientists used a technique from stress engineering, which is often used for the mass fabrication of micro- and nanostructures. On an elastically mismatched structure, which consists of a stiff layer on a compliant surface, a variety of buckling occurs. In this example, the scientists heated a mixture of SiO and Ag2O to prepare the silver-cored, silicon-dioxide-shelled microstructures onto substrates at 1270 K, a temperature above the melting point of silver but below that of silicon dioxide, in order to achieve the proper elasticity difference in the subsequent cooling process.

The thermal stress caused the shell to become unstable, and when the conical-shaped structures were cooled, spherules grew on the most stressed sites. The spherules revealed Fibonacci spiral patterns, sometimes growing in the sinister form and sometimes in the dexter form. The scientists couldn’t determine why one form rather than the other would emerge. While the chirality is considered random in plants, Li, Ji and Cao suggested that a geometrical factor may tip the balance.

“Modeled as the least energy configuration on a confining conical support, one element more or less may reverse the chirality,” said Cao. “We are seeking help on this question from simulations.”

The scientists experimented with different shaped structures, and found that only conical shapes produced Fibonacci spirals with definite chirality. On the other hand, spheres produced triangular patterns, while conical shapes with irregularities (such as the shape of a strawberry) produced an ‘X-pattern’. Still, the scientists noted that conical surfaces do not have to be perfect to produce Fibonacci spirals, which may explain the common occurrence in nature.

“The least energy configuration for particles is dependent on the geometry of the space in which the particles are confined—just look at the thorn bundles of various cacti,” Cao explained. “I only know that the Fibonacci spiral patterns are not the least energy pattern for a sphere (try to imagine a football) or flat plane (we can make Fibonacci spirals on street pavement, but it is not self-assembled). However, Fibonacci spirals do appear on conical receptacles in nature. I conjecture that Fibonacci spirals are the least energy configuration on conical supports, but I cannot prove it.”

For plants, Cao’s conjecture is supported even in very old literature, although scientists have yet to discover a mathematical proof. Cao explained why this proof is so difficult.

“The patterns on a sphere are now referred to as the Thomson problem, which has been generalized as the Generalized Riesz Problem,” Cao said. “There is no general method to find the least energy configuration for a given confining geometry, and the numerical solution costs enormous time of both the computers and the scientists. Even worse, it is difficult to make oneself believe that the least energy solution he finds is really the global minimum. And numerical solutions would never be accepted as proof.”

Cao added that there are many similar “embarrassments” in physics. He explained how physicists know that the body-centered cubic lattice is the least energy configuration of atoms for many metals, which is also currently unproven.

“We do wish that one day this will be proven,” he said. ”You know, as David Hilbert once said, ‘Physics is too difficult for physicists.’ In my opinion, a proof of problems of this kind needs new mathematics.”

Citations: Li, Chaorong, Ji, Ailing, and Cao, Zexian. “Stressed Fibonacci spiral patterns of definite chirality.” Applied Physics Letters 90, 164102 (2007).

Li, Chaorong, Zhang, Xiaona, and Cao, Zexian. “Triangular and Fibonacci number patterns driven by stress on core/shell microstructures.” Science 309, 909 (2005).

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celestial music

Giant pipe organ in the solar atmosphere

Astronomers have found that the atmosphere of the Sun plays a kind of heavenly music. The magnetic field in the outer regions (the corona) of our nearest star forms loops that carry waves and behave rather like a musical instrument.
In a talk on Thursday 19 April at the Royal Astronomical Society National Astronomy Meeting in Preston, Dr Youra Taroyan and Professor Robert von Fay-Siebenburgen of the Solar Physics and Space Plasma Research Centre (SP2RC), University of Sheffield will explain the origin of these magnetic sound waves. They will present a series of animations and sound files that demonstrate how these dramatic events appear and fade away rapidly.

In recent years scientists have worked hard to better explain and predict the dynamic behaviour of the Sun. For example, missions like STEREO and Hinode watch as material is ejected towards the Earth, events which are controlled by the solar magnetic field.

In their research, led by Professor von Fay-Siebenburgen, SP2RC scientists combined observations with new theoretical models to study the magnetic sound waves that are set up along loops in the corona. “These loops can be up to 100 million kilometres long and guide waves and oscillations in a similar way to a pipe organ.” - says Dr Taroyan

The acoustic waves can be extremely powerful and reach amplitudes of tens of kilometres per second. Professor von Fay-Siebenburgen adds, “we found that the waves are often generated at the base of the magnetic pipes by enormous explosions known as micro-flares. These release energy equivalent to millions of hydrogen bombs. After each micro-flare, sound booms are rapidly excited inside the magnetic pipes before decaying in less than an hour and dissipating in the very hot solar corona.”

Source: Royal Astronomical Society

celestial art

'Van Gogh' simulations give new insight into turbulent stars

'Van Gogh' simulations give new insight into turbulent stars
Stunning simulations that give a multi-dimensional glimpse into the interior of stars show that material bubbling around the convection zone induces a rich spectrum of internal gravity waves in the stable layers above and below.
The swirling simulations, which are reminiscent of Van Gogh’s star paintings, show the interior of a star during the Asymptotic Giant Branch (AGB) phase, the final stage of a low- and intermediate-mass star’s life before it becomes a white dwarf. It is during this phase that many of the heavier elements, such as carbon and sodium, are created inside the stars.

The core of the star is surrounded by shells of helium and hydrogen, in which nuclear fusion periodically switches off and on in a process called Helium-Shell Flashes, triggering increased bursts of mixing and heating. The models focus on the turbulent convection zone in the helium shell and show how it interacts with the stable, stratified layers above and below.

Dr Falk Herwig said, “Until recently we’ve only had one-dimensional models of the interior. That’s very different from being able to work out in detail and 2 or 3 dimensions what’s going on deep inside and seeing how the different layers interact. The three-dimensional simulation is so complicated that we have been using as much computing time as some of the large cosmological simulations. This is the first time that we’ve been able to see and actually measure the internal gravity waves that are caused by the convective motions in the unstable layer.”

The simulations show that there is minimal penetration of material into adjoining layers from the convection zone but gravitational mode oscillations with a dominating horizontal velocity component, induced at the convective boundaries do cause some of mixing across layers.

Helium-shell flash convection is dominated by large convective cells that are centered in the lower half of the convection zone. The animations show entropy or temperature fluctuations which start off as small bumps at the bottom of the convection zone and expand upwards, developing mushroom-like instabilities that merge into large-scale features. The stable regions above and below are filled with horizontal waves, which are induced almost as soon as the convection plumes start to grow. This means that the gravity waves are not created by the plumes actually hitting the boundaries, but are induced by the build-up of pressure above the plumes.

Source: RAS

Weekend Project: Make your own hood ornament - Lifehacker

Weekend Project: Make your own hood ornament - Lifehacker

The Eye of the Beholder

ScienceDaily: Eye Diseases Gave Great Painters Different Vision Of Their Work

Eye Diseases Gave Great Painters Different Vision Of Their Work

Michael Marmor, MD, wanted to know what it was like to see through the eyes of an artist. Literally. After writing two books on the topic of artists and eye disease, the Stanford University School of Medicine ophthalmologist decided to go one step further and create images that would show how artists with eye disease actually saw their world and their canvases.
Marmor blurred the image in "Woman Drying Her Hair" to a visual acuity of 20/300 to replicate what Degas might have seen. (Credit: Michael Marmor)
Combining computer simulation with his own medical knowledge, Marmor has recreated images of some of the masterpieces of the French impressionistic painters Claude Monet and Edgar Degas who continued to work while they struggled with cataracts and retinal disease.
The results are striking. In Marmor's simulated versions of how the painters would most likely have seen their work, Degas' later paintings of nude bathers become so blurry it's difficult to see any of the artist's brush strokes. Monet's later paintings of the lily pond and the Japanese bridge at Giverny, when adjusted to reflect the typical symptoms of cataracts, appear dark and muddied. The artist's signature vibrant colors are muted, replaced by browns and yellows.
"These simulations may lead one to question whether the artists intended these late works to look exactly as they do," said Marmor who has long had interest in both the mechanics of vision and the vision of artists. "The fact is that these artists weren't painting in this manner totally for artistic reasons."
Degas and Monet were both founders of the Impressionist era, and their artistic styles were well formed before their eye disease affected their vision. But their paintings grew significantly more abstract in later life as, coincidentally, their eye problems increased.
"Contemporaries of both have noted that their late works were strangely coarse or garish and seemed out of character to the finer works that these artists had produced over the years," Marmor wrote in a paper titled "Ophthalmology and Art: Simulation of Monet's Cataracts and Degas' Retinal Disease" that was published in the December issue of the Archives of Ophthalmology.
It's well-known that such artists as Monet, Degas, Rembrandt, Mary Cassatt and Georgia O'Keefe all reached their heights of artistic vision while facing a decline in their ocular vision. Marmor chose to focus on Degas and Monet for these simulations because both artists suffered from eye disease that was well-documented in historical records, journals and medical histories. Degas had retinal eye disease that frustrated him for the last 50 years of his long career. Monet complained of cataracts interfering with his ability to see colors for 10 years before he finally underwent surgery to have them removed.
"We understand better from these simulations what Degas and Monet struggled with as vision failed," Marmor said.
Over the past 32 years, the Harvard-educated physician has published 200-plus scientific articles on the science of eye disease while at the same time writing about famous artists and how eye disease may have affected their artwork. He authored one book, Degas Through His Own Eyes, and co-authored another, The Eye of the Artist, with James G. Ravin.
"As an ophthalmologist, I'm fascinated with the visual components of art," said Marmor, whose Stanford home is decorated with pieces of modern art that emphasize optical illusions. His family donated works of art to the Cantor Arts Center at Stanford. "I've also spent years talking to patients about the symptoms of their eye diseases. This was a natural outgrowth of my science and art interests."
One museum curator, Richard Kendall, called Marmor's publications on Degas and Monet "of considerable value to the art historical community."
"I consider him one of the most thoughtful commentators from the scientific community on questions of eyesight among French 19th-century artists,'' said Kendall, who is curator-at-large at the Sterling and Francine Clark Art Institute in Williamstown, Mass.
To create the images of the artists' paintings as seen through their own eyes, Marmor used Adobe Photoshop software. He adjusted the blur and filter settings to what he determined would be the different stages of Degas' and Monet's eye diseases, based on medical expertise and historical research.
Degas suffered failing vision from 1860 to 1910. As his eye disease progressed, his paintings grew increasingly rough. From treating hundreds of patients with retinal disease similar to what Degas suffered, Marmor said, he knows that the shading and contrast of images becomes less defined and blurriness increases as such illness progresses.
"Friends would ask Degas, 'Why are you still painting?'" Marmor wrote in his December paper. "His works in the 1870s were drawn quite precisely with facial detail, careful shading and attention to the folding of ballet costumes and towels." By the 1880s and 1890s, the shading lines and details of the face, hair and clothing of the same subjects became progressively less refined.
"After 1900," Marmor said, "these effects were quite extreme and many pictures seem mere shadows of his customary style."
Monet wrote of his growing frustration with his deteriorating vision, describing how he was forced to memorize where the colors were placed on his palette. In 1914 he wrote in his correspondence that colors no longer had the same intensity. "Reds had begun to look muddy," he wrote. "My painting was getting more and more darkened." He was forced to rely on the labels on the tubes of paint in place of his own vision.
"Like retinal disease, cataracts also blur vision," Marmor said, "but more importantly for a painter like Monet, whose style was based on the use of light and color, they can affect the ability to see colors."
"Monet must have struggled mightily as he looked out into the murky yellow-brown garden and tried to decide what subtle impression to create on canvas," Marmor wrote in the December paper. "Slowly progressive age-related cataracts manifest as yellowing and darkening of the lens. This has a major effect on color perception as well as visual acuity."
After reluctantly submitting to cataract surgery in 1923, Monet returned to his original painting style, even throwing away much of the artwork he'd done during the 10-year period that he had cataracts.
"He just couldn't see the colors," Marmor said. "These simulations show how much his sense of color had been destroyed. Some people say, 'Oh, it's a stylistic change.' Gosh, I don't think so."
Understanding the challenges these artists faced because of eye disease helps further illuminate the accomplishments they achieved despite their disabilities, Marmor said.
"There's some reluctance among people in the art world to look outside the historical or psychological influences on the great artists," Marmor said. "I'm open to debate about what these visual changes might mean stylistically or aesthetically. What is not open to debate is what the artists saw. If you ignore that, you're ignoring facts."

Note: This story has been adapted from a news release issued by Stanford University Medical Center.