Ultie Arts

An artist surrounded by cornfeilds and cows, left with nothing but tea, fighting for dignity.

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Are fractals simple or complicated objects? Or perhaps both?  The beauty and attraction of many fractals stems from their complex and intricate form, with ever more detail becoming apparent under increasing magnification. Yet many fractals depend on a very simple rule, applied over and over again, a process called iteration.

The Mandelbrot set is perhaps the best known example. It is completely determined by the very simple formula  z2 + c, where z and c code points in the plane or on a computer screen in terms of ‘complex numbers’.  If, starting at 0 and repeatedly applying the formula to move from one point to the next, the sequence of points stay ‘close to home’, then c belongs to the Mandelbrot set and is coloured black in the pictures. If, on the other hand, the itinerary rapidly shoots off or ‘escapes’ into the distance, then c lies outside the Mandelbrot set and is coloured according to the rate of escape.

This simple rule is very easily programmed on a computer. Yet the Mandelbrot set is an extraordinarily complex object. It has a prominent cardioid, or heart shape, surrounded by near circular buds, which in turn have smaller buds attached to them. On closer inspection, stars, spirals and sea horses become apparent. Joined to these are many fine hairs on which lie miniature copies of the Mandelbrot set itself, and increased magnification reveals an endless gallery of ever more exotic features.

For its appearance alone, the Mandelbrot set would merely be a fascinating curiosity. But in recent years its remarkable mathematical properties have become enormously significant. Naturally associated with each point c of the Mandelbrot set is another fractal, called a Julia set. If c is in the main cardioid, then the Julia set is a closed loop, if c is in the largest bud, then it is formed by infinitely many loops, meeting systematically in pairs, and so on.  Moreover, the Mandelbrot set is ‘universal’ in that it codes the behaviour of iteration by many formulae other than just z2 + c.

Kenneth Falconer is a mathematician who specializes in Fractal Geometry and related topics. He is author of Fractals: A Very Short Introduction.

Images courtesy of Kenneth Falconer.

(via infinity-imagined)

Actual Artist Charcoalandcowpies Quotes

  • Charcoal: I can make them if I can find someone who'll let me use a blow torch.
  • Charcoal: I only set my hair on fire ONCE
  • Charcoal: I ended up in the ER due to a WOOD CHISEL
  • Charcoal: NOT fire

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