Archive for the ‘multiple critical points’ Tag

Fractal No. 1427   Leave a comment


Mandelbrot at the Edge No. 1

Mandelbrot at the Edge No. 1

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Fractal No. 1422   Leave a comment


Internal Mandelbrot Blue

Internal Mandelbrot Blue

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Fractal No. 1385   Leave a comment


Cubics Mandelbrot and Crescents

Cubics Mandelbrot and Crescents

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Daily Fractal No. 1020   Leave a comment


Cubic and Spirals

Cubic and Spirals

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Daily Fractal No. 1010   Leave a comment


Mandelbrots Cubics Copper and Silver

Mandelbrots Cubics Copper and Silver

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Mandelbrot Division


So far adding critical point support to some of the formulae in Saturn and Titan, for the next release, has been straight forward. The next formula “Mandelbrot Division” is somewhat trickier. The formula features two parts that match a standard Mandelbrot with powers that are not necessarily 2, one part is divided into the other, in addition there are two parameters added to the result. All parameters in the new version of Saturn can be a number or a number modified by multiplying or dividing the location in the complex plane, strictly speaking the “Mandelbrot parts” aren’t Mandelbrot unless the parameter is modified, so Mandelbrot Division is only a loose description but it is convenient as a name.

The current formula is as follows:

z = (zα + β)/(zγ + δ) + ε + ζ

To ensure that the critical points can be found a new version of the formula “Mandelbrot Division IP” will be added where IP indicates Integer Power.

z = (zA + α)/(zB + β) + γ + δ

The critical points are found as usual by solving f'(z) = 0, γ and δ simply disappear leaving the Mandelbrot parts to be differentiated using the quotient rule:

(f(z)/g(z))’ = (f'(z)g(z) – g'(z)f(z))/g(z)2 = 0

Multiplying both side by g(z)2 simplifies things so critical points are the solutions of the following:

f'(z)g(z) – g'(z)f(z) = 0

The Mandelbrot parts form the definitions of f(z) and g(z):

f(z) = zA + α
g(z) = zB + β

Their differentials are:

f'(z) = AzA-1
g'(z) = BzB-1

So the formula to solve is:

(AzA-1)(zB + β) – (BzB-1)(zA + α) = 0

which simplifies to:

(A – B)zA+B-1 + βAzA-1 – αBzB-1 = 0

When the values of A and B are known the smallest power can be factored out and the other powers reduced accordingly, as a power of z has been factored out it follows that Mandelbrot Division IP always has a critical point of zero. The other critical points are found by solving the remaining polynomial which will be handled by Saturn and Titan.

This formula hasn’t yet been implemented so I can only show some pictures using the current Mandelbrot Division formula where the initial value is set to zero (a critical point).

Mandelbrot Features

Example 1 – Mandelbrot Features

Example 1 - parameter settings

Example 1 – parameter summary

Example 2 - Multibrots

Example 2 – Multibrots

Example 2 parameter summary

Example 2 parameter summary

The two Titan summaries show changes from the currently released software the most noticeable of which is the display of colour maps at the bottom.

Daily Fractal No. 997   Leave a comment


Octic Byways No. 22

Octic Byways No. 22

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