HIDING IN PLAIN SIGHT How long is the of Singapore? Almost infinite, according to theory. What started as an intellectual exercise by 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 — keep your eyes peeled and nature. rachel sullivan explores you'll see it in some truly strange places

the mathematics of the world corbis (opposite), nasa photos:

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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 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 . 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 . 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. , 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 set, created with a program known as of every nook and cranny either, but he cauliflower, but smaller. And then you cut

corona renderer : Ondrej Karlí P H OTO 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 cooled chamber. This discovery would allow spiral shapes scientists to study snowflakes more closely of galaxies, to wfla k e) under controlled conditions, helping them to understand how the crystals form. the beat of the (sn o aya His 1954 book, Snow Crystals: Natural human heart and Artificial, has been out of print for years, but as SnowCrystal.com (a website branches to small ones in the trees that created by a professor of physics at the make up the forest — and so on, even down

aya); u k ichir o na aya); California Institute of of Technology, in the to the cellular level. United States) puts it: "Nakaya's book offers “There are two types of fractals,” a superb look at a scientific investigation explains Dr Richard Taylor, professor of which begins with almost nothing, and physics, psychology, and art, and director proceeds through systematic observation of the Materials Institute at the towards an accurate description of a University of Oregon in the United States, fascinating natural phenomenon." who is currently developing fractal energy o rbis (main, u k ichir na disease detectors circuits in retinal implants and solar cells. s: c Fractals are being used as a non-invasive “One is exact, where a pattern is P H OTO marker of the progress of diseases like repeated precisely at different scales such Alzheimer’s. Scientists recognise that the as the or 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 insight into the insidious disease. was created, it is remarkably easy.”

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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 while salt deposits s: k appear bright white P H OTO

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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. Micolich, Taylor’s colleague and an associate that he created, plotting the resulting “Then in 1979 I found a book by Benoit professor with the University of New South number geometrically and then plugging Mandelbrot, describing the math behind Wales in Australia. “They represent a way the number back into the equation, and fractals and containing some intriguing of getting a large surface area to exchange iterating the process thousands of times. pictures of fractally-generated landscapes,” resources — in this case oxygen and carbon This 1979 discovery was only made he recalls. “I chewed over this for a couple dioxide — with a small volume cost.” possible because Mandelbrot had access to of months and came up with a family of Micolich, who works on nanoelectronics, the processing power of computers at IBM, algorithms to generate fractals to create says this principle is being used to create where he worked at the time. Since that natural-looking animated landscapes.” ultra-efficient batteries that maximise the time, images of the Mandelbrot set have Natural landscapes have a lot of size of the anode/cathode interface without appeared everywhere, from clothing and randomness, he explains. If you make taking up a lot of space. Elsewhere, fractal car stickers to Cambridge crop circles. an image of a static object it has no antennas are widely used in mobile phones, randomness, so to make it appear realistic to maximise the amount of surface area that you need to add randomness. can receive a signal. Chemical engineers “Computers can generate random too, are applying their understanding numbers by scrambling a set of numbers of the fractal nature of combustion and then giving them back. You can do this repeatedly and use these numbers to plot points on a plane (surface) that will defy "If you want predictability and create a picture,” he a big area to says. “But when you move the camera, or the image is viewed from another angle it transfer lots doesn’t make sense, because when you said of stuff and you wanted random, you got random.” When creating artificial landscapes or you don't want other natural-looking images, consistent to contribute randomness is required, Carpenter adds. The Koch snowflake “To create a natural-looking image you volume because First described in 1904 by Swedish zoom in very closely on an image and then of cost, using a mathematician Niels Fabian Helge von add new bits of randomness in the cracks in Koch, the shape and complexity of the Koch the image using fractal geometry.” fractal design snowflake allegedly inspired Mandelbrot in Using his algorithm, Carpenter made is a great way his quest to measure the British coastline. a film, Vol Libre, showing a flyover of a to achieve it" Using an equilateral triangle as its base, mountain range that appeared remarkably each side of the triangle has another realistic compared with the kinds of processes to reduce the number of volatile triangular kink added in its centre, with animation available then. To put it in gases released when burning coal. And each side of the triangular kink equivalent perspective, Pac-Man, the gold standard in nanoelectronics, fractally branching to one-third of the length of the original animated video game of its time, didn’t patterns of energy dispersal have led to the side. Kinks can be added at smaller and come out until a year later. design of highly efficient energy circuits for smaller scales on each side of the triangle to “It was exciting because I figured out solar cells and nanoscale objects. create a snowflake pattern. how to make movie-quality images at a time “If you want a big area to transfer lots of when animation was limited to crude video stuff and you don’t want to contribute volume game images,” he says. Shortly afterwards because of cost, using a fractal design is a he was hired by Lucasfilm to establish a great way to achieve it,” Micolich says. computer division that would complement

ystems (main) ystems their special effects business. The first major project for the fractal- DIGITAL MAGIC based image rendering software he created Fractals also hide in plain sight in movies as was for the movie Star Trek II: The Wrath well, thanks primarily to Pixar Animation’s of Khan, where Carpenter and his team co-founder and senior scientist Loren created an entire planet. “In the movie, the With both fractals and printing technology Carpenter. Using his knowledge of fractals Genesis device was intended to convert now going three- to develop image synthesis, the fundamental dead planets into living ones. We wanted dimensional, one day 3 DS D irector C reative yttanen, technology that is behind digital film- to realistically simulate this and not have soon you might be able to print educational making, Carpenter's groundbreaking people focusing on bad special effects.” food items, such as fractals work would eventually go on to earn To create the planet, they generated fractal cereal, for him an Academy Award. a and fly-by paths that your child's breakfast Janne K : Janne P H OTO

64 65 DISCOVERY CHANNEL MAGAZINE DISCOVERY CHANNEL MAGAZINE fractals allowed the image to remain realistic even as the viewpoint changed. “The film-makers were so happy with the result that they used the sequence in the next two or three movies,” he laughs. Now, Carpenter says, there are lots of fractals in movies. They are applied to curved surfaces as a way to make rocks or pebbles in a stream more interesting. They're used to create reflectivity, bark on trees, and dirt on the ground. Simulated waves use fractal systems, while lighting programs simulate reflections off wrinkled surfaces. Some software specialises in smoke production, others in making water move realistically. But while extraordinarily realistic animation can be achieved thanks to the use of fractals, generating realistic- the ability to detect people that aren't right is an evolutionary thing, known in animation as the "uncanny valley" looking people that don’t have the audience squirming in their seats is another matter. o c k , feathers) “At Pixar, where up to a million hours are spent on each film, our dinosaurs or imaginary landscapes are rendered so that people accept them as a valid world for the characters,” he says. “However, if you have a cartoon human, people are quite happy to buy the caricature, so we make the skin of rbis (tree); getty images (p o ll o rbis (tree); getty images

our characters look plastic.” ) ; c o But to make a realistic-looking human t is much harder, he says. “It can be achieved at some scales, such as when the person is only a quarter the height of a screen, but when it comes to facial or hand close-ups, S : afp (main ph o not everyone buys it and it becomes creepy — and the last thing we want is for viewers p h oto to move between happy and creepy.” above: jackson pollock's He notes that the human ability to Undulating paths, made in 1947. taylor says pollock detect people that aren’t right is an (above, right) deliberately evolutionary thing, known in the animation imbued his artwork business as the “uncanny valley”. “The with a fractal quality left: a Black Oak tree term was coined by Japanese robotics in the yosemite valley professor Masahiro Mori in 1970, along in the united states, with a theory that says when human with a granite wall as a backdrop. the fractal features look and move almost, but not pattern is plainly visible exactly, like natural human beings, it right: the elaborate causes human observers to feel revulsion.” feathers of a male peacock's tail are a The “valley” refers to the dip in a graph of stunning example of the comfort level of people viewing the not- fractals in nature quite-human features. “The human visual system is drawn to natural-looking objects, and the eye is a natural and powerful fractal detector, much more powerful than a computer,” says

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Taylor. “At an unconscious level, our bodies “When the artwork was finished, I are trained to respond to fractal patterns, noticed it looked a lot like a Pollock painting, which we find soothing.” so once Adam [Micolich] came up with a technique to analyse fractal patterns, we decided to put it to the test by applying it to FRACTAL DETECTION Pollock’s Blue Poles.” So if fractals are everywhere, why did it They found that not only is there an take so long to for them to be recognised? underlying fractal character to Pollock’s “It’s one of those big embarrassments,” he work, the artist knew exactly what he was laughs. “In the mid-1800s mathematicians doing when he created them. “Pollock’s started to generate patterns based on natural style of painting repeats general layering. They called them ‘pathologicals’ and qualities, rather than the intricate iterations thought their patterns superior to anything that characterise fractal imagery like the found in nature, which they thought was Mandelbrot set,” says Taylor. generated by random, haphazard processes.” “He may have started pouring with Taylor notes, “It’s a matter of different intentions, and he may not have conditioning to what we consider to be understood the patterns, but he understood shapes. We are brought up both in maths their quality,” he continues. “Pollock spent and art to recognise patterns that use 10 years refining his technique, and docked Euclidean geometry.” That is to say, shapes his canvases to focus on the central, most like circles, squares, spheres and cubes. fractal part of the work — the fractal quality “We are not consciously trained to look at of the paintings deteriorates towards the messes and see patterns,” he adds. “One of edge of the paintings.” the huge contributions of fractal geometry Taylor’s expertise on Pollock was later is that it has made us reassess exactly what called upon in what became known as the is a pattern, and has given us a language to “Matter Matter”, one of the longest running understand patterns in nature.” and most contentious art controversies of As to why we missed fractals, Micolich the 2000s. A treasure trove of supposed feels this is understandable. “How did we Pollock paintings were found in a storage miss gravity when it was right in front of our locker by a man named Alex Matter, but face?” he asks. “People finally got around their authenticity was almost immediately to studying these natural features, and called into question. Taylor used the Pollock Mandelbrot pulled it all together.” pattern analysis methodology he had While Micolich currently works on created with Micolich to determine whether nanoscale electronic devices, his discovery the newly discovered paintings were of the way electric current travels fractally consistent with the style the artist used in in nanodevices (it branches like a river his other works. Unpopularly, Taylor’s analysis found significant differences between those paintings and established Pollocks. “This was bad news for a lot of people, and controversy over the findings raged for around a year until a group of scientists from Harvard showed that some of the pigment used in the paintings dated from the 1980s. Pollock himself died in 1956, which put the matter to rest.” But just as this scandal was finally going away, another Pollock scandal was arising, with a famous private gallery in New York’s collection of Pollock artworks being found to system) later resulted in unmasking a hoard have been painted by a Chinese artist. wer) of fake Jackson Pollock artworks. “Pollock is a double-edged sword,” o Taylor was Micolich’s PhD supervisor Taylor says. “At a superficial level people are back in the 1990s. By day they worked on convinced that they can paint in his style, nanoelectronics and by night they analysed but there is an amazing equilibrium and o caulifl the compelling patterns and organic shapes composition that can’t be replicated, and This Landsat image of Jackson Pollock’s work, a source of fractals play a key role in creating this.” shows the Mississippi River Delta, in the united fascination to Taylor since he was a child. He adds, “These events show that there states. the way the river “There is a strong relationship between is a bright future for computer scientists cuts through land to maths and art.” After completing a PhD in working in the area of art authenticity. A reach the sea looks very similar to the spread of physics, Taylor went to art school, where he scholar might know whether a work is a tree's branches or the pondered the works of “Jack the Dripper”. authentic or not, but without quantitative shape of a lightning bolt “I knew about fractals from the study of evidence might not be able to convince a jury. left: Romanesco cauliflowers, which physics, and once back at art school thought But while it will never replace scholarship, have a naturally fractal a lot about Pollock. The defining moment science, and in this case , can shape, look almost too came when I built a pendulum structure quantify the materials and processes artists pretty to eat filled with paint that was blown about by the use to create their work — and help ensure (r o manesc (main); getty images usgs/esa p h otos: wind, which has its own fractal motion.” that the truth comes to light.”

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