My name is Ingvar Kullberg, born 1951, a dreamer, mainly interested on Mandelbrot and Julia fractals. This is a description of the way to Shangri la or a short story of my path to the Cubic Parameterspace.

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.

Click the pictures to enlarge

	1) The dream of Jacob in Betel.Detail of b-plane in p(z) = z^3 + 2.1z^2 + b with start z = 0, which is the same set as p(z) = z^3 - 1.47z + b with start z = 0.7
	2) Dragon banners in the west. Detail of b-plane in p(z) =z^3 + 1.8z^2 + b with start z = 0, which is the same set as p(z) = z^3 - 1.08z + b with start z = 0.6.
	3) The birth of the mystery in the cave. Another detail of b-plane in p(z) = z^3 + 1.8z^2 + b with start z = 0, which is the same set as p(z) = z^3 - 1.08z + b with start z = 0.6.
	4) The banners of Chaos in sunset. Detail of b-plane in p(z) = z^3 + 1.995z^2 + b with start z= 0, which is the same set as p(z) = z^3 - 1.326675z + b with start z = 0.665.
	5) The shadow of the troll bursts. Detail of b-plane in p(z) = z^3 + 1.996725z^2 + b with start z = 0, which is the same set as p(z) = z^3 - 1.3289702419z + b with start z = 0.665575.

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.

	6) The enlightened mystery in the cave. The same motive as figure 3, however a little bit zoomed out.
	7) The slope of the crystal mountain. A detail from a region near figure 6.
	8) Fractal compass. The a-slice when b is fixed to zero.
	9) Sunset in Mexico I. Detail of an a-slice when b is fixed to -0.73491841782+0i. Start z = a (=pixel)
	10) Sunset in Mexico II. Another detail of an a-slice when b is fixed to -0.73491841782+0i. Start z = a (=pixel)
	11) The cubic troll flirts with his eyes. Detail of b-plane in p(z) = z^3 + 0.96iz^2 + b with start z= 0, which is the same set as p(z) = z^3 + 0.3072z + b with start z = 0.32i.

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-).

	12) Storm over the Trollmountain. Detail of a_imag and b_imag when a_real is fixed to 0.3 and b_real is fixed to 0.
	13) The Temple of the Northern Wind. Detail of a_imag and b_imag when a_real is fixed to 0.4 and b_real is fixed to 0.
	14) Sunset around Midsummar in Sweden. Detail of b_real and b_imag when a_real is fixed to 0.3 and a_imag is fixed to 0.
	15) The Masquerade of the Juvelqueen. Detail of a_imag and b_imag when a_real is fixed to 0.05 and b_real is fixed to 0.
	16) The Burning Darkness. Detail of a_imag and b_real when a_real is fixed to 0.633 and b_imag is fixed to 0.
	17) Endless Way. Detail of a_real and b_imag when a_imag is fixed to 0.1 and b_real is fixed to 0.2.
	18) Mandy's Totem-pile. Detail of b_real and b_imag when a_real is fixed to 0 and a_imag is fixed to 0.7465.
	19) New! Red Mandy Flower. Detail of the previous image.
	20) Moonlight at Loch Ness. Detail of a_imag and b_imag when a_real is fixed to 0.4 and b_real is fixed to 0. The zoom is rotated about 44 degrees to the right.
	21) Trollfog around the Ghost-galaxy. Also detail of a_imag and b_imag when a_real is fixed to 0.4 and b_real is fixed to 0.
	22) The Lotus of my Heart. Detail of b_real and b_imag when a_real is fixed to 0 and b_imag is fixed to 0.9. The Little Mandy is not artificially placed in the head of the bigger Mandy. It's place is naturally there in this part of cubic parameter space.
	23) The barrier bursts. Detail of b_real and b_imag when a_real is fixed to 0.6583 and a_imag is fixed to 0.
	24) It pains when Knobs are bursting. A magnification of the bursting knob in number 23 above, but only the set M+ is displayed.
	25) The Weapon that are burned out by fire. Detail of b_real and b_imag when a_real is fixed to 0.73 and a_imag is fixed to 0. Only M- is displayed.
	26) Against new horizons. Detail of a_imag and b_imag when a_real is fixed to 0.1 and b_real is fixed to 0.1. Only M- is displayed, the view is entirely inside M+.
	27) Stalactities in the cave. Detail of a_imag and b_imag when a_real is fixed to 0.05 and b_real is fixed to 0.
	28) Attack from the fourth dimension. This is a 3D-section of Cubic Parameter Space, the axis b_imag pointing into the fourth dimension, the 3D-slice is performed at b_imag = 1. The red set is a part of M+, the grey set of M-. The motive is somewhat rotated.
	29) Dark Flames from my Sword. This is a magnification of "3) The birth of the mystery in the cave".

30) New! Twillight over the Hills of Farytales. Detail of a_imag and b_real when a_real is fixed to 0.1 and b_imag is fixed to 0,1.

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

I myself on deviantART

Ingvar Kullberg
SWEDEN

Email-me

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