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As a matter of fact, one can easily be surprised how frequent they actually are. When one looks around in nature searching for fractals there is no scarcity of them.
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The "look like" or "resemble" of its parts does not necessarily mean "identical", it is enough if the parts of the fractal are or look just "similar" to the overall object the parts can therefore also have (random) variations as long as the resemblance with the overall object is still clearly recognisable (it is actually very difficult to exactly define the concept of a fractal without using complex maths, so we leave it here at that and just use the above intuitive verbal description of what a fractal is even Mandelbrot himself had difficulties to exactly define fractals and has changed his definition several times over his lifetime). This defining property of fractals is called " scale invariance". Instead of neglecting these "rough" patterns of many objects in nature he believed that they are representing some essential features and mechanisms worth studying in themselves.Ī fractal is any object (not necessarily a concrete physical object found in nature fractals can also be abstract objects or just mathematical sets or constructs) that can be seen or understood as consisting of parts that resemble the shape and form of the overall object itself just on different, smaller scales, or in other words: a fractal object consists of parts that look like or at least look similar to the object itself just on a smaller scale. Mandelbrot saw structure and recurring patterns in real life objects with rough edges and surfaces that few researchers had noticed before or even studied in detail. We know and study them in standard mathematics and Euclidean geometry in the idealised forms of straight lines, triangles, circles, 3D polygons etc that we all know from elementary math and school. Mandelbrot at the time and throughout his scientific career studied with new mathematical methods the "roughness" and real, non "smooth" and non idealised shapes and contours of real life objects of nature such as everyday rough surfaces of objects (like the barks of trees), boarders and coastlines of countries and islands, the shape of mountain ridges, the formation of river beds and river deltas, all sorts of plants and flowers and even the shape of graphs representing the fluctuations of stock prices.Ĭontrary to most mathematicians of his time he did not see these shapes and forms as mishaps of nature or bad quality versions of "ideal" perfect shapes like the Platonists since Plato do. With these books Mandelbrot laid the foundations for the new mathematical science of Fractal Geometry.
![mandelbulb 3d chromosome mandelbulb 3d chromosome](https://lauragraceweldon.files.wordpress.com/2012/06/fractal-love1.jpg)
His first popular science book about fractals was published in English in 1977 and called "Fractals: Form, Chance and Dimension" and his second and most popular book - and still worth and a recommended reading today - was called "The Fractal Geometry of Nature" published in 1982. I nevertheless hope you enjoy the read - maybe on a rainy day when you have nothing better to do -)įractals can be found pretty much everywhere in nature ! However, fractals as we know and study them today have only become noticed and popular with the general public after the publication of two books by the Polish mathematical genius Benoit Mandelbrot (1924 - 2010 see picture below) who first used and introduced the term "fractals". Even though I cover a lot of ground and topics, the content must remain a bit superficial as some of the underlying scientific background and theories would be very complex and too difficult to explain here. Please note: this is an introductory and overview article. The second part will focus more on fractal processes (rather than spatial fractal structures), fractal time- series, fractals and computational complexity and a further fractal analysis of our brains, language, music, oscillations and resonance, our perception and our thinking processes. I will published the second part of this article early next week. In this first part here I explain the basic concepts and give plenty of examples of fractals as they can be found and are studied in mathematics, physics, complex systems, chaos theory, quantum theory, nature in general, in the arts, in evolution theory and (molecular and quantum) biology and especially in our bodies and brains. I have split the article in two parts because of the length and the time it would take to read the whole article at once. This is the first part of two popular science overview articles I have written about fractals.