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Fractals - Beautiful Mathematics

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Fractals are mathematically generated shapes which have some special properties. The most common types of fractal, which you may have encountered before on snazzy postcards or on the Internet, are the Julia sets and the Mandelbrot set. These are both named after the mathematicians who discovered them, Gaston Julia (1893 - 1978) and Benoit Mandelbrot (1924-2010).

The word fractal stands for 'fractional dimensions' because a fractal does not have an integer (whole number, eg 1, 2, 56, 12 but not 1.63 or 1/2) number of dimensions. To try and clarify this a bit, recall that a line is one-dimensional (1-D), a plane is 2-D and a cube is 3-D. A fractal line, however, can be 1.675-D or 1.0032-D. If this sounds a bit odd, that's because it is. Basically it's a mathematician's way of describing how crinkly a line is.

The origins of fractals lie in the work of Lewis Fry Richardson (1881 - 1953), who asked the seemingly simple question, 'How long is the coastline of Britain?'. If you measure this quantity from a globe of the Earth, with a poorly printed outline of Britain, you might get the answer of a few thousand kilometres. If you measure it from an accurate map with a higher resolution, you would get a greater answer because of all the extra crinkles that show up with a more precise line. However, if you took a metre rule and walked around Britain measuring the coast, the answer arrived at would be bigger again, because a metre stick at that scale would fit into a lot more of the crinkly bits than showed up on the map. If you used a 10cm ruler, again your answer would be bigger, and so on ad infinitum. This leads us to conclude that the coastline of Britain has an infinite length - it is a fractal. The edge of any fractal is in fact infinite, and no matter how small a piece of the line you examine is, it will also be infinite. In effect, since you can make a crinkle as tiny as you like, you can fit an infinite number of crinkles into a tiny space.

The Extremely Complex Bit

Fractals are generated by iterations (repetitions) of a simple formula - for a Mandelbrot fractal z = z2 + c, where z and c are complex numbers. A complex number is a number made by adding a real number (any normal number you can think of is a real number, eg 1, 2.342, pi, -34.232323) to an imaginary number (a real number multiplied by the square root of minus one, this equals i). z begins at 0 + 0i (basically zero) and c is given by the complex plane mapped1 to the screen, so to generate the pixel in the very middle of the screen, c = 0 + 0i. The initial z is squared, and c is added, and again, and again, until the magnitude of z goes above a certain number - generally about four gives good results. The number of iterations required for this to happen is taken as the colour of the pixel. If z never goes above the value, the pixel is given the value 0.

Fractals were originally regarded as nothing but a mathematical curiosity, but now they are being used in computer generated imagery and in image compression technology. Fractals are at the heart of chaos theory, which tries to describe how a tiny initial change in conditions can produce entirely different end results. It is interesting to note how pixels close to each other on a fractal image can have completely different colours. Chaos theory can be used to describe the motions of planets or analyse population changes, so fractals are at the heart of a lot of modern science.

1Either of the two parts into which a cone is divided by the vertex.

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