Of the pictures below #1 - 5 have been produced on my Macintosh IIcx, using Fractal Magic from Sintar Software. There Macintosh version is a commercial version of Dave Platt's MandelZot, an excellent freeware that can be downloaded from the homepage of Dave Platt.I've used morph polynomials, an extension module to this program, to make these five pictures below from cubic parameter space. For the first four I have made 1280x960 resolution small prints (A4, 29.7x21 cm) printed on CLC 800, CLC 1000, which I am selling in Sweden for 60:- including documentation in Swedish. To all of my pictures I have made customizised colour sets with the powerful editor in Dave Platt's program. Here on the net the pictures are shown in 640x480 resolution. Click the picture to magnify.
Short about drawing pictures of cubic parameter space
Any cubic polynomial, according to Branner and Hubbard, can be
parametrised as: p(z) = z^3 - 3a^2 z + b, the coefficient before z
written as -3a^2 in order to make the critical points z = +-a. As there
are two complex parameters, a and b, the cubic parameter space is a
four dimensional hyper space that has to be visualisized in 3- or
2-dimensional slices. The analogy to the Mandelbrot set for cubic
polynomials, also called cubic connectedness locus, therefore
is the set of all complex parameters, a and b, for which the
corresponding Julia sets are connected, i e both critical points, +a
and -a, in the dynamical plane belongs to the filled in Julia sets and
thus have limited orbits under
iteration.
In the above mentioned extension module one is enabled to put different real coefficients before z, z^2, z^3, z^4 and z^5, and then study the parameter space. The problem is that MandelZot always use z = 0 as input and that input is not a critical point if we write the cubic polynomial centered as above. However I got the idea to parametrise the cubic polynomials non centered under the form: p(z) =z^3 + kz^2 + b with the critical points z = 0 and z = -2k/3 as input, which would produce the same set(s) as for the polynomial p(z) = z^3 - 3a^2 z + b with the critical points z = +-a despite from the location in the parameter space. (There is one problem: If you put b = 0 and study the k-section using z = 0 as input, every k-parameter would result in bounded orbits).
This theory I got verified when I received abFractal of Les
Pieniazek, 16314 Clearcrest Dr Houston, Tx 77059. This is the only
application allowing me to study some 2-dimensional slices (the
b-sections for fixed a, even complex) of this four dimensional
monster in a correct manner. I say "correct" because in this
application one can freely choice the first input for z. Every
published picture made by morph polynomials, is verified to be correct using abFractal. However this application among others
have the disadvantage not allowing more than 253 as max iteration.
What is the relation between a and k in the above polynomials? Suddenly I just knew that the relation must be k = 3a, which I verified using my computer. It took me some days to verify it mathematically. It runs with the assumption that the distance between the two critical points must be the same in both cases, i e:
|-2k/3 - 0| = |-a - (+a)|
2k/3 = 2a
k = 3a
With the help of this "discovery" (strong for me) I have made the pictures below, using the above mentioned extension module.
The following pictures 6 - 11 are produced
with on a PC with the application Flarium24 by
Stephen Fergusson
This application was the first one that made it possible for me to investigate a_real, a_imag for different fixed b-real, b_imag. This thanks to the formula-editor and the fact that the program normally uses z = c as z_0.
In figures 6 - 8 one-periodic components of the set are coloured with inside colours denoting the number of iterations required for taking the critical orbit near the fixed point.
Since the above images are produced, a dear friend has written a wonderful module for
Ultra fractal
specially written for intensive studies of cubic parameter space.
The following images are produced with his module. In most of these images
two layers are used, one with start z = +a (displaying M+), and the other with start z = -a (displaying M-).
30) New! Twillight over the Hills of Farytales. |
Regards,
Ingvar
Click here to see
my twenty pictures from the quadratics and the two pictures from my
Compass Formula. more info about my Compass Formula can be read in 27) Compasses in my Chaotic series (see below).
And if you are interested here is my humble Cubic Tutorial.
Chaotic series of fractal articles
Homepage of Dave Platt | Homepage of Rollo Silver
Homepage of Art Matrix
Regards:
Ingvar Kullberg
SWEDEN
Unique visitors since 2010-08-16
ŠIngvar Kullberg
First day for this page on the Internet was 1997-11-11
Uppdated 2008-09-29