January 22, 2001. Latest updated February 2003.
Hi Fractal People!
(Note: Because Stig has made a new version of his Cubic Parameterspace called
Cubic Parameterspace2 included in his spr-module, I’ve modified the formulas in
the first group to this sub-module. They are called MilnorAB2, SorensenCubic2
etc. The resulting images are the same as those obtained by the older set of sub-
modules. The older sub-modules are still in my “ik” for compatibility, but I
strongly recommend everyone to use the new versions for new images. The
attached parameter-files mentioned below use the new sub-modules.)
In order to help you to get started with my ik-module, I’ve supplied a set of
parameter-files, described below. In those I’ve made certain setups for
magnification, period-check on/of etc. I’ve turned the period-check “of” where
artifacts seem to appear. The sub-modules, all dealing with cubics, are divided into
two groups. In the first group I have replaced the standard formula, z -> p(z) = z^3
- 3a^2 z + b with some other variants used by the great mathematicians. In order to
get these variants I’ve modified Stig’s sub-module Cubic Parameterspace. In the
second group the standard-formula is employed, but other methods than making
regular slices of the four dimensional cubic monster are used. I don’t know the
mathematical meaning, but I suspect them to be some kind of “projection-
methods”. Most of them I have received them from Stephen C Ferguson, who has
received them from Professor Holger Jaenisch. Not being a programmer I'm guilty
many thanks to Stig Pettersson, who kindly have helped me with this module,
especially by attaching Switch Julia to the second group. Thanks also to Stephen C
Ferguson who sent me the source-code for the “projection-methods”. Now a brief
description of my sub-modules (Hint: The sign "^" means 'power of', for example
z^3 means 'the third power of z', i e z*z*z. The sign “_” means ‘index’, and
"sqrt(z)" means 'the square-rot of z)’.
First group. Variations of the cubic formula:
1) Cubic Parameterspace: First I have supplied a startup-file for Stig’s “Cubic
Parameterspace” just for reference. Here the standard formula, z -> z^3 - 3a^2 z +
b is dealt with. I suppose most of you have his sp-module, as well as spr (earlier
sp2), in the formula-folder of UF. If not it can be downloaded from:
http://w1.111.telia.com/~u11108896/ambrosid/plugins.html
In all startup-files two layers are used, named M+ and M- each having it’s own
color-gradient. M- is the set were “z” is initialized to the other critical point, in this
case z = -a. In this startup-file (b_real, b_imag) is plotted and (a_real, a_imag) are
fixed to zero, which gives the standard Cubic Mandelbrot set. If you fix a_real to
0.57735 by typing this value to “z-axis” for both M+ and M- you will obtain the
same figure as figure 7 (last Figure) in my “Cubic Tutorial”:
http://user.tninet.se/~cim027f/CubTut/cubictut.html
If you don’t have read this tutorial yet, it can be suitable to do so now, as all
obscure terms in this GetStarted are explained there. Note: Every change done in
dialog-boxes must be done for both M+ and M-. In the following startup-files
(a_real, a_imag) are plotted as default.
2) MilnorAB: Here I've modified Stig's sub-module in order to study cubic
parameter space in the way of Professor John Milnor in his paper "Remarks on
iterated cubic maps" published in 1991. From the well-known standard-formula
z -> p(z) = z^3 - 3a^2 z + b, Milnor have done the substitution A = a^2 and B =
b^2 in order to study what he calls "the Moduli Space". That means that he iterates
z -> z^3 -3Az + sqrt(B) with the critical points z = +sqrt(A) and z = -sqrt(A)
instead of z = +a and z = -a. The critical points are obtained by putting the
derivative to zero.
3) SorensenCubic: The Danish mathematician Dan Sörensen has written some
small special applications for Macintosh for drawing fractal for scientific studies.
For Cubics he, besides the standard formula (z -> z^3 - 3a^2 z + b), also uses the
formula: z -> z^3 + az^2 + bz. Here the critical points becomes z = -a/3 +
sqrt(a^2/9 - b/3) and z = -a/3 - sqrt(a^2/9 - b/3). The parent fractals have large
range, so I’ve put the magnification to 0.5 in order to see the whole slices of the
set. A strange thing when looking at slices of (a_real, a_imag) is that the whole of
the right half of the plane belongs to M+, and the whole of the left half of the plane
belongs to M-. Therefor it's suitable to use two layers, one for each subset, as been
done in my parameter-files. In fact that's almost always suitable to do so when
drawing 2D-slices of cubic parameter space.
4) KullbergCubic: Using the non-centered parametrization p(z) = z^3 + az^2 + b
you get the critical points z = 0 and z = -2a/3. The story of this can be read at:
http://user.tninet.se/~cim027f/frholmes/cubic.html
Here, by modifying the submodule of Stig, all 6 perpendicular systems of planes
can be studied and also with respect to the other critical point. When displaying
(a_real, a_imag), when "b" is fixed to zero, and z is initialized to zero, you get a
completely black screen. That's natural because for every "a" you get the orbit 0 ->
0 ->0. However if you initialize z to the other critical point "-2a/3" you obtain a
“CCAP-shape” (however more than twice as big). Displaying (b_real, b_imag)
gives the same images as (b_real, b_imag) in the standard-formula. The
coordinates, however, are different. See the above URL.
5) DevaneyCubic: In his book "A First Course in Chaotic Dynamical Systems"
Professor Robert Devaney makes the parametrization p(z) = z^3 + az + b, which
gives the critical points z = +sqrt(a/3)i and z = -sqrt(a/3)i.
6) BrannerSpecialCubic: In a paper "The iteration of cubic polynomials. Part II:
Patterns and parapatterns" (Acta Mathematica 69: 3 - 4, 229 - 325), the two great
mathematicians Bodil Branner and John Hamal Hubbard in one place (page 237)
make the alternative parametrization p(z) = (b/4 - a/4)(z^3 - 3z) + (a + b)/2 in
order to prove a certain statement, which I don’t understand a bit of. However I
decided to include the function in my module. The two critical points turns out to
be z = +1 and z = -1. Besides normal features of pictures of cubic parameter space,
there are two slices, (a_real, a_imag) with “b” fixed to +1+0i, and (b_real, b_imag)
with “a” fixed to -1+0i (M+ and M- coalesces in these slices) which have very
interesting properties. I leave to the fractal-exploring people to make their own
investigation.
7) EpsteinCubic (NEW): In the paper GEOGRAPHY OF THE CUBIC
CONNECTEDNESS LOCUS I: INTERTWINING SURGERY the great
mathematicians Adam Epstein and Michael Yampolsky besides the standard
iteration-formula use the formula z -> a(z^3 – 3z) + b. Critical points are z = +1
and z = -1. In the special case if you plott the b-plane when “a” is fixed to zero,
you obtain a completely black screen (natural since you for every initializing of “z”
from the second iteration obtain the orbit b -> b -> b etc).
The second group. Other methods of displaying 2D images of the standard-
formula:
1) CBAP and 2) CCAP: These variants of cubics occur in some of the applications
of Ferguson. He has obtained the formulas from a Professor Holger Jaenisch who,
I suspect has obtained them from the very big mathematicians on the field of
iteration of cubic polynomials. The reason for this assumption is that I've seen
details of both in a slide series from Art Matrix in the early nineties. In one of the
scenes in the two hours video-show "MANDELBROT SETS and JULIA SETS"
from the above company, there is a deep zoom sequence in CBAP. Asking
Ferguson for the source code of these variants, he displayed the CBAP code along
with some of his images on abpf. Later he sent this and the below formulas to me
by email (thanks Steve!). The quasi-code runs as:
init:
t=#pixel
a=(t^2+1)/3t
b=2a^3+(t^2-2)/3t
z=-a
loop:
z = z^3 - 3a^2 z + b
By an accident, when playing around with the above expression, I received CCAP.
This was done simply by deleting "+1" and "-2" from the above! Being of a to large
scale it received the correct size by deleting "3". NOTE: If you do that on the
unmodified CBAP above, you get another fractal. After abbreviation the quasi-
code of CCAP runs as:
init:
a=#pixel
b=2a^3+a
z=-a
loop:
z = z^3 - 3a^2 z + b
From “Lecture 12” (see 6 below) I’ve now learned that “b” is selected so that p(a)
= a. This means that a^3 – 3a^3 + b = a which makes b = 2a^3 + a. The fact that
the critical point z = +a always goes to a fix-point, and thus M+ covers the whole
plane, explains why the secondary decorations attached to the copies of the
Mandelbrot sets in M- have the shape of 1-periodic Julia sets with parameter
values picked from the “center” of the Mandelbrot set.
The shape of CCAP also occurs in SörensenCubic and KullbergCubic in (a_real,
a_imag) when “b” is fixed to zero. However the size in these two cases are
between 2 – 3 times as big.
3) CFAP and 4) CGAP: Also these formulas come from Professor Holger
Jaenisch. The shape of CFAP also occurs in Milnor AB in (A_real, B_imag),
SorensenCubic in (b_real, b_imag), and DevaneyCubic and EpsteinCubic in
(a_real, a_imag) when intersecting origo. However the size in the two last cases
are twice as big. There are also differences of the Julia sets, especially between
CFAP and the other three. I leave to the diligent fractal explorers to make there
own investigations. The quasi-code of CFAP runs as:
init:
a=#pixel
b=2a^3-2a
z=-a
loop:
z = z^3 - 3a^2 z + b
and the quasi-code of CGAP runs as:
init:
a=#pixel
b=2a^3+1
z=-a
loop:
z = z^3 - 3a^2 z + b
5) SteveCubic: This cubic formula occurs in Flarium24 as #31 and is created by
Stephen Ferguson himself. The quasi-code runs as:
init:
t=#pixel
a=(t^3-1)/3t
b=2a^3-2a
z=-a
loop:
z = z^3 - 3a^2 z + b
The origin and meaning of these formulas I have no understanding of. I suspect
they are some kind of "projections" rather than slices. An important thing regarding
Julia sets from all the above methods as far as I’ve seen is that they all can be
obtained from ordinary slices of cubic parameter space.
6) HomerCubic: This displaying variant I received from UNIVERSITY OF
ROCHESTER Mathematics Lecture 12. Today the link is corrupt. The resulting
fractal turned out to be the same as one in the scenes in the video-show
"MANDELBROT SETS and JULIA SETS" from Art Matrix. I named it
HomerCubic because Homer Wilson Smith together with Jane Elizabeth Staller are
the editors of this video-show. The C-locus of the set is also displayed by Rudy
Rucker, who named it “the Rudy set” in his site:
http://www.mathcs.sjsu.edu/faculty/rucker/cubic_mandel.htm
The quasi-code runs as:
init:
a=#pixel
b=a
z=+a (The layer M+)
z=-a (The layer M-)
loop:
z = z^3 - 3a^2 z + b
7) MysticCubic: This variant is also received from “Lecture 12” The parameter “b”
is chosen so that p(-a) = +a. This means that (-a)^3 + 3a^3 + b = +a which makes b
= a - 2a^3. This also means that only one critical orbit needs to be tested, and both
subsets coalesce. The fractal has the same properties as the last mentioned slices in
BrannerSpecialCubics. That’s the reason for naming it ” MysticCubic”. The quasi-
code runs as:
init:
a=#pixel
b=a-2a^3
z=a
loop:
z = z^3 - 3a^2 z + b
Non-cubic formulas:
Article15: The iteration-formula is z -> z^2 + pz + c. You can vary “p” (z-
coefficient) and the Starting point (the value to which “z” is initialized) and draw
the c-plane. If you put the “Starting point” to minus half the z- coefficient, you
always come up with the standard Mandelbrot set. Otherwise you come up with a
so-called “Perturbed M set”. All this is dealt with in Article15 in the “Chaotic
series”, at the moment under uploading at FAME:
http://www.wack.ch/fame/artists/kod101.html
NOTE: This sub-module is written only for pedagogical reasons and does not
contribute any new forms that can not be brought out from the standard
Mandelbrot formula in Ultra Fractal.
Multicorns: This is a generalizing of the tricorn-formula z -> conj(z)^2 + c to the
general z -> conj(z)^d + c with free choice of the exponent “d”. If z = x + iy,
conj(z) = x – iy. I got the idea to this generalization from a note in a paper. In the
attached startup-parameter file Diff-Bailout (see below) is enabled to denote 1-
periodic component which otherwise would be black
Compasses: This module performs the iteration: z -> z^d – d a^(d-1) z, the critical
point being z = a, and the “a-plane” the parameter plane. This formula is
constructed in connection with Article26, Compasses, on some fractal mail groups,
where the cases where the exponent ”d” is settled to 2, 3, 4, ……etc is discussed.
The title “Compasses” refers on the fact that for d = 3 and higher integers give rise
to compass-like fractals. In fact the exponent “d” in this formula can be set to any
complex number. Here also Diff-bailout is enabled in the attached startup-
parameter file.
Deformed Mandekbrot: Draws the four-dimensional parameter-space for
quadratics parametrizised as z -> z^2 – 2az +b, “z” initiated to the critical point
“a”. This sub-module is written for Article28 in the Chaotic series in which it is
shown that one parameter is enough for quadratics. The interface is the same as in
Cubicparameterspace2.
Diff-Bailout:
To the formulas in the second group (CBAP etc) and to the new non-cubic
formulas the feature Diff-Bailout is adopted Cubic Parameterspace2. When “Diff
Bailout” is enabled, 1-periodic components are shown with inside-coloring, the
colors display the number of iterations required to take the variable “z” in close
proximity to a fix point. NOTE: In order to avoid artifacts, turn the peroid check
“off”, and set the bailout to at least 10 000 000. A high iteration-number, 1000 or
so, is also recommendable.
The periods of hyperbolic components can even be seen using Stig’s new inside
color routine “Period”. Components with different periods are colored in different
solid colors. If one wish to now the period of a certain period, for example period
1, type “1” under “Period” and click “enable” under “Research” and components
with other periodicity will show ordinary the set-color (usually black).
Fixed bug:
In the old versions based on the old module “Cubic Parameterspace” there is one
bug of little interest. It’s concerned with the first group as well as with original
itself. If you have a parameter-plane flipped or rotated, the Switch-Julias is the
same as if the parameter-plane was not flipped or rotated. This bug is fixed in the
new versions. Suggestions and comments are always welcome. Enjoy!!!
About Julias:
You shall not take the default-Julias too seriously. I’ve changed the default seed
from (0, 0) to other values in cases where (0, 0) would give rise to a completely
black screen. In fact the best method to produce nice Julia sets is to use “Switch
mode” in a parameter-plane (non-Julia fractal) and click when you see a nice shape.
Regards,
Ingvar Kullberg
www.come.to/kullberg
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