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34 35 fAscinAtinG frActAls selfconsistency to the nth degree There are many natural phenomena, perhaps the greater part, about which the term symmetrical seems to have little relevance. The amorphous shapes of clouds, the rugged contours of mountains, the turbulence of streams, the patchiness of lichen, etc., especially taken together, create a distinct impression of confused irregularity. But there are consistencies in all of these things, the uncovering of which has greatly extended the notion of selfsimilarity, and of symmetry itself. Many natural formations, even though they may appear highly complex and irregular, possess a recognisable statistical selfsimilarity. This means that they look the same across a range of different scales, and the degree of their fractality can be accurately measured. There is, moreover, a converse application of this notion that highly complex phenomena may have a hidden order, namely, that relatively simple formulae can create highly involved figures. The renowned Mandelbrot set background opp. is probably the best known and most complex example of this effect. In fact, many organic structures exhibit the fractal properties of self similarity animal circulatory systems, for instance. The branching, systems of blood vessels, which repeat on an everreducing scale, allow the most efficient circulation of blood to every part of the body. In mathematics many kinds of fractals are unlimited by scale and can, in theory, go on to infinity, but this is seldom the case in the real world, especially in living creatures where the rule is fitness for purpose. Bloodvessels do not reduce indefinitely, any more than the whorls withinwhorls of the fractal cauliflower extend to infinity. Nature uses fractal geometry where it is advantageous. Sierpinski gasket Koch snowflake Sierpinski carpetcube Sierpinski hexagon Fractals are linked to the enormous advances in computer science and Chaos Theory, but their geometry has a history of its own. The above forms, dating from the early 20th century, were originally seen as mathematical curiosities that demonstrated the mingling of finite spaces and infinite boundaries
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