Let's start off with the number line. This is a conceptual line on which all positive and negative numbers lie. So one point on it is marked zero, and a point to the right of that is marked one. An equal distance away is two, and so on. On the left of zero are minus one, minus two, and so on.

Two and a half, unsurprisingly, is located halfway between two and three. Even numbers not expressed as fractions can be approximately shown on the line. The square root of two (approximately 1.4) is between one and two.

The number line extends infinitely in both directions.

Now, let's talk about square roots. The square root of a number is another number that multiplied with itself yields the original number. For example, the square root of four is two, because two times two equals four. Most square roots of integer numbers aren't integer numbers themselves - the roots of two, three or five aren't. But you can locate them all on the number line.

But what about the square root of minus four? What number, multiplied by itself, yields minus four? A moment's thought shows there can be no such number: Any positive number multiplied by itself yields a positive number, and any negative number multiplied by itself also yields a positive number.

Given that no ordinary (no real) number will do the trick, we have to make up a new set of numbers which do. These are called the imaginary numbers.

Just as the "base unit" of real numbers is called 1, the base unit of the imaginary numbers is called i.

You can have multiple is: 2i, 3i, -9.38i. You can also mix real and imaginary numbers: 2 + 3i. These are called the complex numbers.

And if you multiply two imaginary numbers, you get a real, negative number. (That's the whole point.)

3i x 2i = -6

Indeed, you can do pretty much any kind of maths with complex numbers, though it's a bit more, er, complex:

(3 + i) x (5 - 2i) = 15 - 6i + 5i + 2 = 17 - i

But how can we describe complex numbers on the number line? They don't really fit anywhere, but we can extend the number line to accomodate them.

First, we can make up another number line for the imaginary numbers:

Then we put that number line at a right angle with the original one. They can meet at zero, because zero times i is the same as zero times one. These two lines define a surface on which we can locate any complex number. This surface is called the complex plane.

This is the point in this explanation where we get to the fractals: the Mandelbrot set is a set of complex numbers. And if you mark all the numbers in the set on the complex plane, you get this familiar picture:

By set of numbers, by the way, I do just mean "a bunch of numbers". This is a set of numbers: { 1, 2, 3, 4, 5, 99 }. As is this one: { 5, -99, 3i }. Another set of numbers is the set of all even numbers bigger than four. Of course, there are infinitely many numbers in the set, but it's not necessary to write out a set to define it - all you need is a test to see if a given number is in a set.

The test can be simple, like "all numbers smaller than three", complicated, like "all prime numbers the sum of whose digits is a square" or even "all numbers mentioned in Moby Dick".

The test to see if a given number is in the Mandelbrot set is as follows: take the number, and square it. Then add the original number to the result. Then square the result of that. Add the original number again, and so on.

If the result of this process gets bigger and bigger the more you square-and-add, the original number is not in the set. Otherwise, it is. Note that "bigger" can also mean "really big negative number like -1223432443" or "really big imaginary number like 329239823i".

To put it into more mathematical terms, if the function tends towards positive or negative infinity, the number is not in the set.

Of course, we can only do this square-and-add process a finite number of times, so we can never be quite sure for some numbers. It's easy enough to see that 3 is not in the Mandelbrot set, because if you repeatedly square-and-add 3, it gets pretty massive pretty soon:

3² + 3 = 12
12² + 3 = 147
147² + 3 = 21612
21612² + 3 = 467078550
etc.

Equally, you can probably guess that -1 is going to be in the set, because:

-1² + -1 = 0
0² + -1 = -1
-1² + -1 = 0
etc.

But it gets less clear with other numbers, and some of them may teeter on the brink quite long before suddenly getting large, while other may forever stay small. And the closer you look, the more difficult it gets to tell those two kinds of number apart. You end up doing the square-and-add thousands of times and still can't be sure.

In fact, it turns out that there seems to be no end to this uncertainty. For every number you find that isn't quite in the set, there is another number that is very nearly the same as the first, but which never grows big and hence is included.

This means that there is no way of saying "this is where the border of the Mandelbrot set is". The closer you look, the more detailed the border becomes. As a result of this, the circumference of the Mandelbrot set is considered to be infinite: the more precisely you measure it, the bigger it becomes.

While this is already pretty trippy, add to this what's called self-similarity. As you zoom into the set for a closer look, you will notice smaller shapes that look similar but not-quite-identical to the shape of the original set. And next to them, yet smaller ones. The Mandelbrot shape, and other shapes, are endlessly repeated among the spirals and tides of the set.

Now go forth and explore!

If you zoom in far enough, you will find a little message saying "maximum zoom reached". This isn't due to the set, but rather due to the limitations of the way your computer stores numbers.

You'll have noticed that there is more than one fractal to choose from. Apart from the "classic" Mandelbrot set, there are also some minor variations:

The second through fifth fractals work much like the Mandelbrot set, but instead of squaring the number in the test, you take it to the power of respectively three, four or five. (Note the interesting correspondence between exponents and symmetry.)

The sixth fractal is called the Burning Ship, discovered by Michael Michelitsch and Otto E. Rössler. Its test uses squaring like the Mandelbrot set, but makes the real and imaginary parts of the number positive before squaring.

Finally, the seventh fractal is the Tricorn or Mandelbar Set, whose test takes the complex conjugate of the number before squaring. The complex conjugate of a number is its real component minus its imaginary component - so the complex conjugate of 3 + i is 3 - i.