Introduction to the Mandelbrot Set

This help article explains briefly what the Mandelbrot Set is and what Julia Sets are. They are mathematically defined; for a more technical definition, see Technical Information.

The Mandelbrot Set

To determine whether a point is in the Mandelbrot Set: Imagine a journey, starting at the centre of the page, the point with co-ordinates (0,0). The journey then moves to the point under consideration, then to another point, then on to another, and so on forever. One of two things will happen. For some points, this journey will remain bounded – that is, it will remain confined within a fairly small area. The other is that it will, after a number of steps, start to head off to infinity – moving further and further away.

If the journey remains bounded for ever, the point is considered to be in the Mandelbrot Set, and is coloured black. If the journey eventually starts to head off to infinity, the point is outside the Mandebrot Set, and is allocated a colour depending upon its “dwell” – that is the number of steps before the journey began to head off to infinity.

The definition of a Julia Set is similar. The difference is this: Before, we started the journey at the point (0,0), and varied the definition of the journey so that the first stop is a different point each time. For a Julia Set, the definition of the journey is the same each time. What we vary is the point where the journey begins.

Obviously, no computer can go on computing the journey for a point literally forever. Therefore we have to set a value for a “Maximum Dwell” – how many steps of the journey will we calculate before we give up: if the journey is still bounded by this point we conclude it is inside the Set. (Mandelbrot Explorer sets this value appropriately for your automatically, unless you override the setting.)

For a more technical definition, see Technical Information.

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