FRACTALS HIDING IN PLAIN SIGHT HOW LONG IS THE COAST OF SINGAPORE? ALMOST INFINITE, ACCORDING TO FRACTAL THEORY. WHAT STARTED AS AN INTELLECTUAL EXERCISE BY BENOIT MANDELBROT OPENED THE DOOR TO A WHOLE NEW BRANCH OF MATHEMATICS, WHETHER IT'S GLACIER MOVEMENTS (OPPOSITE) AND REVEALED THE ORDER THAT OR TREES, FRACTAL GEOMETRY IS FOUND UNDERLIES ALMOST EVERY PART OF ALMOST EVERYWHERE IN NATURE — KEEP your EYES PEELED AND NASA (OPPOSITE), CORBIS (OPPOSITE), NASA NATURE. RACHEL SULLIVAN EXPLORES you'LL SEE IT IN SOME truly STRANGE places THE MATHEMATICS OF THE WORLD PHOTOS: 56 57 DISCOVERY CHANNEL MAGAZINE DISCOVERY CHANNEL MAGAZINE FRACTALS ACCORDING TO ONE RELIABLE SET OF MEASUREMENTS, Singapore’s cOASTLINE IS 193 KILOMETRES, AUSTRALIA’s is 25,760 KILOMETRES, AND THE UNITED Kingdom’s IS 12,429 KILOMETRES. ANOTHER EQUALLY RESPECTABLE SET OF MEASUREMENTS GIVES SINGAPORE A VALUABLE ADDITIONAL 75 KILOMETRES (FOR A TOTAL OF 268); AUSTRALIA GETS WHOPPING EXTRA 40,770 KILOMETRES TO ITS NAME; AND THE UNITED KINGDOM A MORE MODEST BUT STILL IMPRESSIVE 7,288 KILOMETRES EXTRA. SO WHY THE DISPARITY? t all depends on the unit mountains, trees and many other natural of measurement. If you objects, are made of irregular shapes that measure in lots of 50 endlessly repeat themselves at different kilometres then you get the scales. He found a way to rate their first set of measurements. roughness, and opened the door to a whole But because a land mass new branch of mathematics. has features at scales that Unlike geometrically smooth surfaces range from hundreds of such as one-dimensional lines, two- metres to the nanoscale, dimensional triangles or three-dimensional the smaller the unit of spheres, Mandelbrot found that so-called measurement, the longer fractal shapes, irregular patterns that are the coastline will appear reproduced infinitely at all scales to produce to be. So even though natural phenomena, fall somewhere logically a coastline has between two and three dimensions. The a finite length, from a rougher the surface, he discovered, the mathematical perspective higher its fractal dimension. it is infinite. Speaking at a TED talk shortly before This paradox left his death in 2010, the Yale mathematician, cartographers scratching a citizen of France and the US, used the their heads, until Polish- example of a romanesco cauliflower to born mathematician Benoit illustrate the concept of fractals, and Mandelbrot brought a new approach to the of self-similarity. Self-similarity is one table. In attempting to accurately measure of fractals’ most important underlying the coastline of Great Britain, Mandelbrot principles, and arises from endless found something that had apparently eluded repetition of a simple pattern. conscious human minds for millennia — the “A cauliflower is both very complicated THIS IS A close-UP OF A numerical order that is in fact underlying and very simple at the same time,” he said. MANDELBULB, A THREE- DIMENSIONAL analogue nature’s apparent chaos. “If you cut one of the florets with a sharp OF THE MANDELBrot Mandelbrot couldn’t measure the length knife and look at it, you see the whole ONDREJ KARLÍ SET, created WITH A : PROGRAM Known AS of every nook and cranny either, but he cauliflower, but smaller. And then you cut OTO H CORONA RENDERER P did recognise that coastlines, like clouds, again, and again, and again… and each time 58 59 DISCOVERY CHANNEL MAGAZINE DISCOVERY CHANNEL MAGAZINE FRACTALS you see smaller cauliflowers. Humans have always recognised that some shapes have RECREating this peculiar property, where each part is like the whole, only smaller.” natURE'S ARTWORK In geometric terms a coastline is a fractal, which remains similar-looking at all scales. Like the distance around a lake, he noted, "The concept of the length of a coastline should be straightforward. It is published in many places — but is in fact a complete fallacy because each time you get closer, the longer the distance becomes.” PRACTICAL DESIGN Fractal shapes are found everywhere in nature and at all scales, from the rings of Saturn and the spiral shapes of galaxies, to the beat of the human heart; from the branching of our blood vessels, nervous system and DNA, to the structure of Japanese physicist Ukichiro Nakaya (below) mountain ranges, river deltas and seashells. has been credited as being the first person Electrical impulses travel across wires in to create an artificial snowflake in the lab. a fractal pattern, whether they are high- Nakaya's obituary in the journal Arctic tensile wires or nanowires. And researchers stated, "At the time Nakaya took over the in the US state of Arizona have found that new department of physics [in Japan's the ratio of large trees to small ones in a Hokkaido University], there was a minimum forest exactly matches the ratio of large of equipment and few funds available for research. But he did have a microscope and FRACTAL SHAPES an unlimited number of natural snow crystals during the long winters." ARE FOUND In total, Nakaya captured around 3,000 EVERYWHERE IN images of natural snow crystals, and later proposed a general classification system for NATURE AND AT snowflakes. Afterwards, he began trying to ALL SCALES, FROM recreate nature's artwork in the laboratory. In 1933, he succeeded, growing the THE RINGS OF world's first artificial snowflake on a single SATURN AND THE rabbit hair, by circulating water vapour in a E) K cooled chamber. This discovery would allow SPIRAL SHAPES scientists to study snowflakes more closely WFLA OF GALAXIES, TO O under controlled conditions, helping them to understand how the crystals form. THE BEAT OF THE (SN AYA K His 1954 book, Snow Crystals: Natural NA HUMAN HEART O and Artificial, has been out of print for years, but as SnowCrystal.com (a website ICHIR branches to small ones in the trees that K created by a professor of physics at the make up the forest — and so on, even down AYA); U AYA); California Institute of of Technology, in the to the cellular level. K United States) puts it: "Nakaya's book offers NA “There are two types of fractals,” O a superb look at a scientific investigation explains Dr Richard Taylor, professor of ICHIR K which begins with almost nothing, and physics, psychology, and art, and director proceeds through systematic observation of the Materials Science Institute at the towards an accurate description of a University of Oregon in the United States, RBIS (MAIN, U fascinating natural phenomenon." who is currently developing fractal energy O DISEasE DETECTORS C circuits in retinal implants and solar cells. S: OTO Fractals are being used as a non-invasive “One is exact, where a pattern is H P marker of the progress of diseases like repeated precisely at different scales such Alzheimer’s. Scientists recognise that the as the Mandelbrot set or Koch snowflake ABOVE: A RESIN cast OF central nervous system and the activities it (see page 65),” he says. “The other is a THE HUMAN lungs AND controls have an intrinsic fractal structure, BRONCHIAL TREE. THE natural fractal, which is more subtle. In with similar patterns, whether monitored AIRWays OF each lobe OF natural fractals the pattern doesn’t repeat THE lung have BEEN cast in minutes or hours. These activity patterns precisely but the general quality, known IN A DIFFERENT colour. THIS change with age and disease, and Harvard LARGE NETWORK OF AIRWays as statistical self-similarity, is repeated at scientists have recently discovered that as GIVES THE lungs A HUGE smaller and smaller scales.” surface AREA for OXygen Alzheimer’s disease progresses in a patient, Taylor notes, “Nature in the raw can be to ENTER THE blood AND the disruption to their fractal activity carbon DIOXIDE to EXIT mind-bogglingly complex, but when you look patterns increase, giving doctors additional beyond and think about how the complexity insight into the insidious disease. was created, it is remarkably easy.” 60 61 DISCOVERY CHANNEL MAGAZINE DISCOVERY CHANNEL MAGAZINE FRACTALS SOUTH KOREA'S KOMPSAT-2 SATELLITE CAPTURED BOTH THIS IMAGE, FEATURING ROLLING HILLS OF FARMLAND IN THE US STATE OF WASHINGTON, AND THE IMAGE OPPOSITE, WHICH PICTURES THE SAND SEAS OF THE NAMIB DESERT IN SOUTH AFRICA. THE BLUE AND WHITE AREA IS THE DRY RIVER BED OF THE TSAUCHAB. BLACK DOTS OF VEGETATION ARE CONCENTRATED CLOSE TO THE RIVER’S MAIN ROUTE, ARI/ESA K WHILE SALT DEPOSITS S: APPEAR BRIGHT WHITE OTO H P 62 63 DISCOVERY CHANNEL MAGAZINE DISCOVERY CHANNEL MAGAZINE FRACTALS Nature’s functions, he says, rely on “I started out as a computer programmer taking a simple pattern, and then repeating FAMOUS FRACtals back in the ‘60s when computers were the pattern over and over again. “You need hard to come by,” he explains, adding that only two pieces of information — one that while studying computer science he worked describes the basic pattern, and another at Boeing on rendering algorithms that that describes how to repeat it. This simple MANDELBROT SET showed how concept aircraft might fly. formula allows something as complicated “Everything Boeing puts out has a as a tree or a cloud to be built,” he explains. As well as measuring natural shapes, mountain in the background. It implies “Out of such complex, rough objects you get Mandelbrot is perhaps best known for the aircraft is safe because it is flying over amazing functionality that smooth-shaped popularising a shape that had never mountainous terrain,” he says. To display objects don’t have.” before been seen. He generated the the concept aircraft in classic Boeing style, “It is no coincidence that lungs and trees eponymous pattern by feeding complex he wanted to make pictures of aeroplanes share a similar shape,” adds Dr Adam numbers into a very simple equation passing in front of mountains.