This post refers to Saturn and Neptune, they are programs for exploring fractals and are available from the download page.
Octic fractals are generated using an eighth order polynomial, the eighth power Mandelbrot is a special case where all coefficients except that for z8 are zero. It has a single critical point at zero in common with all Mandelbrots defined using the following general formula:
z = zn + c
where n is an integer greater than or equal to 2.
The eighth power Mandelbrot can be found in the list of Saturn fractal types under the name M08 Mandelbrot. It looks like this:
Saturn also has a fractal type called Octic which is a full polynomial, its formula is:
z = αz8 + βz7 + γz6 + δz5 + εz4 + ζz3 + ηz2 + θz + ι
For use with Mandelbrot algorithm at least one of the parameters (or coefficients) must be the location in the complex plane (usually denoted as c) for convenience ι is set to c. When using the Mandelbrot algorithm the initial value of z is set to a critical point, other values can be used and resulting pictures are “perturbed”. So what are the critical points of an “Octic”? Critical points are the roots of the first derivative of the fractal formula.
So, for Octic:
f(z) = αz8 + βz7 + γz6 + δz5 + εz4 + ζz3 + ηz2 + θz + c
the first derivative is:
f'(z) = 8αz7 + 7βz6 + 6γz5 + 5δz4 + 4εz3 + 3ζz2 + 2ηz + θ
The critical points are the roots of:
f'(z) = 0
I mentioned earlier that any of the coefficients of the polynomial can be the location in the complex plane, in which case, the critical points have to be calculated for every location. Neither Saturn nor Neptune can do this which is why ι was set to c. Saturn only allows a number or the location in the complex plane to be used as the initial value for z. So what value should be used? It is possible to zero out some of the terms so that some critical points can be found relatively easily but that precludes the use of all the terms, solving f'(z) = 0 where all the terms of the polynomial are present isn’t easy, fortunately you don’t have to work them out as Neptune can do it for you. Neptune was specifically designed to produce pictures of fractals with multiple critical points, its version of Octic is MC 220.127.116.11.18.104.22.168.
To start the exploration of Octic fractals all parameters except ι are set to 1, i.e.:
f(z) = z8 + z7 + z6 + z5 + z4 + z3 + z2 + z + c
Neptune produces this picture:
Saturn can only produce pictures using one critical value at a time, the z0 value to be used can be found on Neptune’s settings window:
Using the first and sixth critical points Saturn produces:
Zooming into an area at the left hand size of the sixth critical point picture standard second power Mandelbrot islands can be found.
There are no eighth power Mandelbrot islands (multibrots) as found in the first picture of this post.
Saturn can also generate Octic Julias, the Julia algorithm uses c as the initial value where all parameters have fixed values. Here is an example Julia using the Julia form of the formula used for the pictures above:
There is an enormous variety of Octic “Mandelbrot sets” here are two examples using different coefficients:
Where all terms of the polynomial are present only Octic Mandelbrot sets will contain second power Mandelbrot islands. Zeroing out various terms other than z8 and c results in sets that contain both Mandelbrot and cubic Mandelbrot islands. Higher order Mandelbrot islands are also found, I don’t know whether more than two types of Mandelbrot islands can be present in the same set, over the course of the next parts I intend to find out.
Neptune will be exclusively used from now on, so no more Julias. Neptune has 9 version of the Octic formula with the number of critical points varying from 7 to 2:
- MC 8.1 z = αz8 + βz + c
- MC 8.2 z = αz8 + βz2 + c
- MC 8.3 z = αz8 + βz3 + c
- MC 8.4 z = αz8 + βz4 + c
- MC 8.5 z = αz8 + βz5 + c
- MC 8.6 z = αz8 + βz6 + c
- MC 8.7 z = αz8 + βz7 + c
- MC 8.7.6 z = αz8 + βz7 + γz6 + c
- MC 22.214.171.124.126.96.36.199 z = αz8 + βz7 + γz6 + δz5 + εz4 + ζz3 + ηz2 + θz + c
Part 2 will focus on MC 8.1.