## The Mandelbrot Set

The picture you see behind this text is the Mandelbrot set, probably the most famous of all fractals. This odd shaped image is created with an extremely simple formula: `Z = Z * Z + C`.

All of the pictures in the gallery labelled as 'Mandelbrot' can be found somewhere inside the backdrop image you see behind this text.

Honest.

We've changed the colours used, and zoomed in a lot in a lot of different places.

Odd facts about the Mandelbrot set:

• The area of the Mandelbrot set is unknown, but it's fairly small.
• The length of the border is known - it's infinite!
• The barnacle covered pear shape that you see occurs an infinite number of times in the Mandelbrot set. Rotated, distorted and shrunken, but quite recognizeable.
• All of the black areas of the Mandelbrot set are connected together.
• Every band of colour around the Mandelbrot set (not shown on this image) goes all the way around, without breaking, and without crossing any other colour bands. Think about that when looking at some of the more complex areas!
• The Mandelbrot set can be used as a very inefficient way to calculate PI.
• The Mandelbrot set is named after its discoverer, Benoit B. Mandelbrot.
• Without computers, the Mandelbrot set would be invisible. Even a very low-resolution image of the Mandelbrot set requires millions of calculations.
• The Mandelbrot set has infinite detail - you can keep zooming in forever. Fractal eXtreme 'only' lets you zoom in to two thousand digits of precision - more than you are ever likely to need.
How can such a simple formula create such incredible complexity? Because it is used in a feedback loop. The equation is calculated dozens, or even millions of times for each pixel. Each time through the loop, the result of one calculation is used as the input for the next calculation. It is this feedback loop that gives the Mandelbrot set, and many other fractals, their complex behaviour.

The speed of modern micro-processors, plus some very careful assembly language coding (the Mandelbrot calcs alone took several weeks of coding and optimization) allow Fractal eXtreme to explore the Mandelbrot set quickly and easily.