The previous set of posts on the Compasses fractal back in 2011 did not cover positive integer powers greater than 2. The Compasses formula has changed slightly, there are now three parameters instead of two, the location in the complex plane, c, was incorporated directly in the original formula. The current version of Saturn allows any parameter to be set to c or -c so the original formula can be reproduced.

The formula is:

z = z^{?} – ??^{? – 1}z + ?

With ? = 2, ? = c and ? = 0 the formula becomes:

z = z^{2} – 2cz

To find the critical point f'(z) = 0 needs to be solved:

f(z) = z^{2} + 2cz

f'(z) = 2z – 2c =…

so there is one critical point c, the location in the complex plane. The initial value of z is set…

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