| Mathematicians did a great disservice to us mere mortals by naming the map on which they plot graphics, the complex plane. It puts us off! There is nothing complex about it - it is simply like a map with map references. To reach a point on a map, two numbers are needed - the distance across the top and the distance down the side. Just so with the Complex Plane. These references are then known as complex numbers because each contains two numbers and not one! Similarly, instead of x across and y up and down, they call the ones across 'real' and the ones up and down 'imaginary'! Now although there is a bit more to it, like square roots of -1, that's it in a nutshell! Most plotting is done by a process of 'iteration'. That is simply putting a number through a formula and then taking the result and putting it through the formula again. | ||
| This shows the 'Complex' plane with it's 'real' plot going across and 'imaginary' up and down. The complex number 0.5 -1.0 is shown plotted just like a map reference. |
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Escape time fractals, which include the majority of those shown here, are plotted by taking a screen pixel, applying the formula and seeing what happens. In the case of the Mandelbrot set the fomula used is Z(n+1)=Z(n^2)+c. Z(n+1) is the new plot. Z is the first number which starts in the centre at 0,0 and c is the address of the screen pixel being tested. The result is inserted into the formula and the next point ascertained and so on. These points are called the orbit of that screen address. If the result becomes bigger than the bailout number (2 in the case of the Mandelbrot set), then the plot has gone outside the boundary of the set and will eventually end up at infinity. If the plot stays inside the bailout number, it is inside the set. Some will always stay inside, some will always go to infinity. The original screen pixel is now coloured in 2 different colours depending on the result and gives the typical pattern of the Mandelbrot lakes and bays. |
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| This shows the result of plotting the set as above in 2 colours. |
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If a different colour is given to each point depending on how many iterations were performed before it 'escaped', then the picture becomes much more complex and often eerily beautiful. A limit has to be set on how long the feed back of numbers continues to see if the plot 'escapes' or the process can go on for ever. This is the number of iterations, which can be set in the software. The most complex part of the plot is around the 'shore' of the lakes and here, the higher the iterations performed, the more detailed will be the result.
By enlarging small parts of the set, and replotting the points on the screen, an huge world opens up of swirls, seahorses, bottomless pits, etc, like those show in the plots section here. It seems unbelievable that such a simple formula can produce such infinite complexity. |
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