The Quadratic fractal is similar to the standard Mandelbrot formula with the addition of an extra term:
z = αz2 + βz + γ
The default values are α = 1, β = 1 and γ = c (i.e. the location in the complex plane).
The formula is the first I implemented in an experimental program called Venus written in C using Gtk+, it was then included in Mars (C++ and gtkmm) and then in Saturn (also C++ and gtkmm) and finally in the current version (C++ and Qt). At the time I did not know about critical points so the initial value was set to zero just like the standard Mandelbrot.
The default image looks like this:
Once I had learnt about critical points I applied them the formula, the critical points are the solutions of f’(z) = 0.
f(z) = αzsup>2 + βz + γ
f’(z) = 2αz + β = 0
so, there is only one critical point at -β/2α and is -0.5 for the default Quadratic. The resulting picture is:
The original picture turns out to be just a ‘perturbed’ Mandelbrot where the initial value is not a critical point. The first picture can be produced using the standard Mandelbrot formula and using 0.5 or -0.5 for the initial values of z. Using the Quadratic formula: no matter what values are used for α and β a standard Mandelbrot will appear, only its size and position will change.
I had thought a standard Mandelbrot would be produced when α and β were substituted with the location in the complex plane either together or separately. The difficulty was that when only one of them was substituted the resulting critical point is dependent on c e.g. α = c, β = 1, critical point = -0.5/c. The current version of Saturn can only set the initial value of z to a number or to the position in the transformed or untransformed complex plane.
I’m currently working on the next version of Saturn which will address the problem of setting the initial value of z to critical points dependent on c. In addition complex parameters such as α and β will be modified by the location in the complex plane instead of being substituted with c or -c. Complex parameters can be multiplied by c or divided by c so where c was substituted the equivalent value will be (1 + 0i) multiplied by c. In Saturn the symbols for multiplication and division will be used. The new fractal settings window looks like this:
Note the value for γ. Unused parameter tabs are no longer displayed and there are two new tabs ‘Critical Points’ and ‘Notes’, the new tabs will also be hidden when there are no critical points or notes.
When a critical point or points are dependent on c ‘location dependent’ will be displayed in the critical points tab.
Now that critical points can be set for the Quadratic fractal and its parameters can be multiplied or divided by c exploration of the Quadratic formula can be greatly enhanced. The resulting pictures certainly are NOT copies of the standard Mandelbrot set:
Note the truncated fat Mandelbrot, this can be cured by increasing the limit value, 1600 is used to produce this:
So that’s a brief exploration provided by a preview of new features in Saturn. The new version of Saturn will be released sometime in 2015.
The focus of this part on Octic fractals is on the MC 8.4 formula:
z = αz8 + βz4 + c
The critical points are found by solving f’(z) = 0:
f(z) = αz8 + βz4 + c
f’(z) = 8αz7 + 4βz3 + c
z3(8αz5 + 4α) = 0
so the solutions are 0 and the four fourth roots of -β/2α.
There are five critical points one fewer than MC 8.3. The pictures of each component show that that there are only two different pictures, the pictures for all the non-zero critical point components are identical.
The component parts look like this:
Combined the resulting picture looks like this:
This fractal features M4 and M2 Mandelbrots which are illustrated in the following pictures:
I’ve tried various values for α and β and all result in identical pictures for the non-zero critical points.
The next part will deal with the M8.5 formula.