![]() These fractal attractors lie at the heart of Chaos Theory. Remarkably, no matter the starting point and the order in which the mappings were traversed, the point would quickly be sucked onto a “strange attractor” - the fractal shape - and once there, it would dance around on it forever. Each hop was determined by selecting one of the mappings at random. The Chaos Game took a starting point in space and tracked its motion as it hopped around. The shape of a fractal can be completely captured by a small list of mathematical mappings that describe exactly how the smaller copies are arranged to form the whole fractal.īarnsley’s influential 1988 book Fractals Everywhere contained an algorithm, known as the Chaos Game, that allowed computers to quickly generate any fractal shape from its known mappings. Though the geometry of fractal shapes is infinitely complex, a third trait of fractals is that their complexity arises from very simple core definitions. – Benoît Mandelbröt, 1924-2010Īs Mandelbröt was putting fractals under the microscope, the British mathematician Michael Barnsley (currently of the Australian National University) was approaching the same objects from a different angle. Complexity from simplicityīottomless wonders spring from simple rules which are repeated without end. Tiny bits of the fractal can look exactly the same as the whole.įar from being a mathematical curiosity, this zoom symmetry can be found everywhere in nature - once you know to look for it.Ī bolt of lightning reveals its zoom symmetry for a split second - each branch resembles a small copy of the whole shape. Incredibly, zooming in on a small region of a fractal leaves you looking at the same shape you started with. The infinite intricacy of fractals permits them a completely new type of symmetry that isn’t found in ordinary shapes. To mathematicians, a symmetry is an action that when applied to a shape will leave it looking (more or less) the same.įor instance, we say that a square has rotational symmetry because there’s no way to tell if a square has been spun around by 90 degrees when you weren’t looking. These computer programs allow you to spot a new kind of symmetry associated with fractals. ![]() Though practically impossible to create by hand, modern interactive applets (such as this one, created by British designer Paul Neave) allow you to create and explore these sets in real time. To choose the right colour for a specific point, we apply a simple movement rule to the point over and over again and watch how long it takes for the point to “escape” off the page. Note how smaller pieces of the set closely resemble the whole. The Mandelbröt set, a famous fractal that can only be drawn by computers. Weierstrass’ shape had irregular details at every possible scale - the first key feature of a fractal shape. No matter how many times the shape was magnified, any glimmer of a smooth line would invariably dissolve into a never-ending cascade of corners, packed ever-more tightly together. To show it could be done, the German mathematician Karl Weierstrass constructed a zig-zag that was so jagged, it was nothing but corners - the ultimate mathematical staccato. If you pick up a pen and doodle a zig-zag, you should end up with a number of sharp corners connected by smooth lines. ![]() In 1861, the discovery of the world’s first fractal sent shockwaves through the mathematical community. – Benoît Mandelbröt, The Fractal Geometry of Nature Many patterns of nature are so irregular and fragmented that, compared with Euclid … Nature exhibits not simply a higher degree but an altogether different level of complexity. How can we describe a fern as a precise mathematical shape? How can we build a mathematical model of this wonderful object? Enter a completely new world of beautiful shapes: a branch of mathematics known as fractal geometry. Such regular shapes are extremely rare in nature. The International Space Station, an engineering wonder whose shape can be modelled by classical Euclidean geometry.
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