The focus of this part on Octic fractals is on the MC 8.3 formula:
z = αz8 + βz3 + c
The critical points are found by solving f'(z) = 0:
f(z) = αz8 + βz3 + c
f'(z) = 8αz7 + 3βz2 + c
z2(8αz5 + 7α) = 0
so the solutions are 0 and the five fifth roots of -7α/8β.
As there are 6 solutions there are 6 critical points one fewer than the critical points for MC 8.1 and MC 8.2.
When α and β both equal 1 and ignoring outer colouring where inner and outer areas overlap the fractal looks like this:
The fractal is made up of 6 component parts, one for each critical point. Pictures of the individual component parts are shown below:
None of the components occupy the same area of any of the other components. Two pairs of pictures can be identified where one of each pair is a mirror image of its counterpart.
The buds on the outline of the fractal resemble the buds of the standard (or M2) Mandelbrot and the cubic (or M3) Mandelbrot. In addition the fractal features M2 and M3 Mandelbrot islands.
Multiple critical point fractals are more interesting when the colouring does not ignore outer colouring where inner and outer areas overlap. When outer colouring is ignored and the inner colouring method is a fixed colour (usually black) the structure of overlapping areas is lost. The following picture shows the components of the fractal merged together and the colours used in each component layer are melded together (in this case all colouring is averaged):
Varying the value of β alters the appearance of the Octic fractal produced by MC 8.3 in the same manner as MC 8.1 and MC 8.2 so the next part will focus on MC 8.4.
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 = zα – αβα – 1z + γ
With α = 2, β = c and γ = 0 the formula becomes:
z = z2 – 2cz
To find the critical point f'(z) = 0 needs to be solved:
f(z) = z2 + 2cz
f'(z) = 2z – 2c = 0
so there is one critical point c, the location in the complex plane. The initial value of z is set to the location in the complex plane, so a different value is used for each point in the fractal’s picture. When I’ve shown how to derive the critical value in other posts the value has been a number which is used for all points in the fractal picture. Currently, Saturn allows the initial value of z to be set to a number, the transformed or untransformed location in the complex plane which is fine in this case but will prove to be inadequate.
The picture produced for α = 2, β = c and γ = 0 is as follows:
Changing the value of γ alters the appearance of the fractal but as γ is not present in f'(z) the critical point is not affected.
Increasing α to 3 produces a different fractal, again using β = c and γ = 0 the critical point can be found:
f(z) = z3 + 3c2z
f'(z) = 3z3 – 3c2 = 0
3z2 – 3c2
obviously the critical point is again c, the location in the complex plane and the fractal produced looks like this:
All looks fine, M2 Mandelbrot type buds and no barren areas, however, change the value of γ and all is not so fine:
A picture of Saturn’s main window shows a barren area highlighted within the rectangle:
Why has this happened? There are two critical points, not one. I have encountered a phenomenon with Neptune where more than one critical point can produce the same image which is the case when γ = 0. Neptune is a program for generating “multiple critical point” fractals, it combines the component parts, one for each critical point, of the fractal into a final image.
To get more than one critical point where 3z2 – 3c2 treat c2 as being 1 multiplied by c2, the square root of 1 is 1 and -1 hence two critical points c and -c. Saturn can’t set z to -c and Neptune can’t calculate critical points dependent on the location in the complex plane which is why Neptune does not include any Compasses formulae.
For higher values of α the critical points are the roots of 1 multiplied by c, so for &alpha = 4 the roots are 1 and two complex values, for α = 5 the roots are 1, -1, i and -i. The roots of 1 are always equally spaced on a unit circle centred at the origin of the complex plane.
The next versions of Saturn & Titan and Neptune & Triton will support critical points dependent on the location in the complex plane. The next part will focus on setting γ to c.
This post is a continuation of the MC 8.2 version of the octic formula. As a reminder this is the formula:
z = αz8 + βz2 + c
Octic fractals that are calculated using polynomials can produce bizarre shapes and bits can separate off the main mass in the form of M2 Mandelbrots (i.e similar to the standard Mandelbrot). The Mandelbrot fragments are usually distorted in some way, the distortion is greater the closer the fragment is to the main mass.
Here are some pictures, mostly of the fractal as whole but with some highlighted regions. The colouring mostly ignores overlapping outer areas. Some of the pictures are coloured taking into account the overlapping areas and those pictures show detail that is obscured by ignoring the overlapping outer areas.
Note: the value for α in all the following pictures is 1.
The smaller the value of β rleative to α the closer the fractal produced is to the standard M8 Mandelbrot. The following picture looks as though it is an M8 Mandelbrot but unlike the M8 Mandelbrot it does not have any M8 Mandelbrot islands. It has M2 Mandelbrot islands and often islands that do not match any of the standard Mandelbrots from M2 to M8.
The same area as above for the standard M8 Mandelbrot showing multiple M8 Mandelbrot islands:
So far the Octic fractals have the buds and islands characteristic of the standard Mandelbrot set. The next part will show a change with the substitution of the βz2 term with a βz3 term, i.e. MC 8.3.