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I think I owe Jules Ruis an apology. I was going by Internet disinformation. Julia sets are not unbounded.

I have sorted it out. Each pixel has its range from the origin and azimuth angle:

With Mandelbrot, THIS is the vector of what we will call the "TUMBLE". According to de Moivre, one doubles the azimuth angle when one squares the complex co-ordinates:


According to de Moivre, one squares the range:


These two operations, the doubling of the angle and the squaring of the range, I shall call the "SPIN".:


Such a "spin drier" does not deliver fractals. The secret is in the tumble, which turns it into a mathematical spin-and-tumble drier. With Mandelbrot, the tumble is of a magnitude and direction parallel with the original pixel position relative to the origin:


This "spin-and-tumble" completes one iteration. The iterations continue for that dot, until the position escapes the 2-unit-radius circle. However, throughout the evaluation of that pixel, the tumble-vector is taken from that pixel:


So the tumble-vector changes from pixel to pixel in the Mandelbrot set. There are, however, infinite numbers of Julia sets - each with its own vector - which have a fixed vector, the same for all pixels. Here is one such Julia set. I have used minus root a half for the real (X= -0.707106781) and plus root one eighth for the imaginary (Y= 0.35355339). This works out as ninety pixels "tumble" to the left and forty-five upwards:


Both systems are BOUNDED. The pixel is scanned sequentially from an array. Each pixel has its own reserved space. There is no contention for space, no vector-graphics (spacial patterns). There is simply an analysis of how many recursions are possible, and then a colouring-in of the pixel within its bounds.

Charles