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## Fractal photo finish

The Mandelbrot set. Image Wolfgang Beyer.

If you're bored with your holiday snaps, then why not turn them
into fractals? A new result by US mathematicians shows that you can
turn any reasonable 2D shape into a fractal, and the fractals involved
are very special too. They are intimately related to the famous
*Mandelbrot set*.

The new result, which was proved by Kathryn A. Lindsey and the late
William P. Thurston, works for any shape whose outline is made up of
a finite collection of
*simple closed curves*. These are continuous loops in the plane
that don't intersect themselves. The fractals involved come from
seeing how points in the plane are moved around by a particular class of mathematical function.

To understand how this works, think about the quadratic function

When you plug into the function, the result,

is again 0, so 0 is what’s called a *fixed point*. The same happens with as

Other values of , however, are more interesting. Take for example. We get

If you now apply the function again using 4 as your new value for you get:

Using 16 as the new value for you get

You can see what happens if you repeat this process, always using the result of your last calculation as the new value for The numbers you get become bigger and bigger, in fact you can make arbitrarily large numbers just by repeating this procedure. The sequence of numbers you get in this way, starting with the number 2, is called the *orbit* of 2 under the function , and we say that the orbit of 2 *escapes to infinity*. In fact, for this particular function the orbit of any number that is bigger than 1 or smaller than -1 escapes to infinity.

But what happens for numbers that lie between -1 and 1, that is, for ? Squaring a number between 0 and 1 always gives you something smaller and positive, so clearly these numbers don’t have orbits that escape to infinity. And since the square of a number between -1 and 0 is a number between 0 and 1, these points don’t have an escaping orbit either. As an example, look at the orbit of

and so on. The numbers get smaller and smaller, they certainly don’t rush off to infinity. So what we’ve just established is that the orbit of all points with or escape to infinity, while the orbits of all other points remain trapped in the interval

You can play a similar game for any function involving powers of , such as

or

These functions are called *polynomials* and their general form is

where are constants. For all such functions there’ll be a selection of points whose orbits escape and a selection of points whose orbits remain trapped within an interval.

The black area is the filled Julia set of a quadratic polynomial. Its outline is the Julia set.

*complex numbers*.

(See the article Unveiling the Mandelbrot set for details.)

Just as before with the number line, a function *f* now gives a partition of the plane into points with
escaping and points with trapped orbits. The set of points with trapped
orbits is called the *filled Julia set* of the function
(after the French mathematician Gaston Julia). And the boundary of the filled Julia set, its outline, is
called simply the *Julia set*. Generally Julia sets of
polynomials are fractals: shapes of infinite complexity. Their crinkly
outline
remains just as crinkly, no matter how closely you zoom in on them. (See this article for more on fractals.)

The new result says that any simple closed curve in the plane can
be approximated by the Julia set of a polynomial — you can match the curve by the Julia set — and that such an
approximation can be made as precise as you like. If you're trying to
approximate a whole collection of simple closed curves, you may need to
use functions that are slightly more complicated. They are called *rational
functions* and are what you get from dividing one polynomial by
another. The collection of curves can then be approximated by parts of
the Julia sets of these functions.

A Julia set in the shape of a cat. To see its crinkliness, look at this larger version. Image: Kathryn Lindsey and William Thurston.

And what's the connection to the Mandelbrot set? It turns out that the points on the plane can be used to label quadratic polynomials: every single quadratic polynomial can be associated to exactly one point in the plane (see Unveiling the Mandelbrot set for details). The points that belong to the Mandelbrot set, coloured black in the picture of the Mandelbrot set above, correspond to quadratic polynomials whose Julia sets consist of just one component, rather than a collection of separate chunks. The Mandelbrot set is itself a fractal and it's pretty universal. It doesn't only come up in connection with quadratic polynomials, but also in connection with many other classes of functions.

A preprint paper of the new result has been published on arxiv.org. Not the easiest way of spicing up your photos perhaps, but one that will delight mathematicians.