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.

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Of all the arts, the novel is the most thoughtful, the closest, and the most personal. It can be about anything, and can take any form or forms it chooses. The novel, like the human species, is now global and the form is still coming to terms with this deep and recent change.
—  From ‘Chapter One: Saying everything’ in Contemporary Fiction: A Very Short Introduction, on Very Short Introductions Online. Free until 24 April 2014.

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Very Short Fact: On this day in 1509, Henry VIII of England marries Katherine of Aragon.

Armed with a papal bull of dispensation, the teenage king married Katherine of Aragon, his brother Arthur’s widow, declaring himself to be deeply in love and sweeping aside objections that she was nearly six years older than he was.

[p. 20, The Tudors: A Very Short Introduction, by John Guy]

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Image: Portrait of a princess(Infanta Catherine of Aragon or Mary Rose Tudor?) by Michel Sittow. Public domain via Wikimedia Commons.

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Very Short Fact: Lord Byron

On this day in 1788, poet Lord Byron was born.

The most famous poet of the 19th century was Lord Byron, a Scottish Calvinist exile from the English high society that he scorned, that bitched about him, but that could not get enough of his charisma.

[p. 145, English Literature: A Very Short Introduction, by Jonathan Bate