Some fractal formulae may have multiple critical points but those critical points produce fewer pictures than the number of critical points. The following:

z_{n+1} = αz_{n}^{2} + βz_{n}^{-2} + c

is just such a formula. Setting both α and β to 1 the critical points are found by solving the following equation:

f'(z) = 2z – 2z^{-3} + c = 0

so

z^{4} = 1

which produces critical points at 1, -1, i and -1.

Neptune, a program for producing multiple critical points, is currently in development and produces this picture.

The fractal is produced by calculating the formula four times one for each critical point which is a waste as only two pictures are possible. The picture for critical points 1 and -1 is:

Critical points 1 and -1

and the picture for critical points i and -i is:

Critical points i and -i

A mechanism in Neptune is required to select the critical points to be used in producing a picture that way two critical points can be used halving the execution time.

The formula has two parameters, changing their values affects the pictures produced. For the next pictures the values of α and β are varied with α and β equal so that the critical points are maintained at 1, -1, i and -i.

α and β = 0.75

α and β = 0.5

α and β = 0.4

α and β = 0.34

As the values of α and β are reduced the bailout value needs to be increased to prevent locations being regarded as escaped when they are in fact captive.

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