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/Ingvar kullberg
Ingvar Kullberg skrev:
> In article 27, Compasses, in my chaotic series of
> fractal articles,
>
> http://klippan.seths.se/fractals/articles
>
> I've made a generalisation of the cubic iteration
> formula, z -> z^3 - 2a^2 z + b to z -> z^d - 2a^(d-1) z
> in order to see what happens when d = 2, 3, 4, 5, etc,
> the a-plane plotted, b is fixed to zero, and z
> initialised to "a", which always is at least one
> of the critical points. More about this can be
> read in the article referred to above. Why this
> formula is called "Compasses" will be obvious
> when looking at the illustrations. I've written the
> formula so you can put "d" as ANY REAL AND COMPLEX
> NUMBER. The results in these later cases are very
> amazing and interesting.
>
> NOW I HAVE made this iteration formula complete,
> in as much I've added the second parameter "b",
> i e we iterate z -> z^d - 2a^(d-1) z + b, the
> parameters (a, b), forming a four dimensional
> hyper space. In this formula, ExtendedCompasses,
> you can plot all 6 system of slices, and even
> make rotations between the planes in the same way
> as in Cubic Parameterspace3, written by my dear
> friend Stig Pettersson. My new formula, included
> in the ik3-folder, as well as the modules of Stig,
> startup-parameter, and pdf-manual can be downloaded
> from:
>
> http://klippan.seths.se/fractals/articles/modules.zip
>
> You will always find copies of the ordinary (quadratic)
> Mandelbrot set surrounded by more or less crazy
> patterns in some areas! Below some illustrations
> (b = 0, so the motives could be drawn by the old
> "Compasses" as well):
>
> http://klippan.seths.se/illustrations/illustrations/LovelySpiral.jpg
>
> The above motive is a detail when the exponent
> d = -4.54545354+1.875778i. The numbers before
> the real and imaginary part were typed down by
> me when I were in some kind of transcendental mode.
> The next motive
>
> http://klippan.seths.se/illustrations/illustrations/LovelyBrot.jpg
>
> is zoomed in at the spot pointed out by the yellow
> arrow in the top of the motive. The below motive
>
>
http://klippan.seths.se/illustrations/illustrations/LovelyAbsenceBrot.jpg
>
>
> is zoomed in at the spot pointed out by the green
> arrow at the bottom of the motive. The four-armed
> star in the middle seems to denote a place where
> there ought to be a minibrot! No filters are used,
> I promise.
>
> The UF-parameter file for the first motive is at
> the very end of this article. Having run this, the
> two following motives can easily be found by zooming
> at the spots denoted by the arrows. Even the un-
> magnified fractal (I call it "parent fractal")
>
> http://klippan.seths.se/lists/Diverse/FeatherSpiral.jpg
>
> can easily be found by simply outzooming.
>
> Play, use a high bailout, and move carefully
> along the non plotted axis'!
>
> The next article, I hope it will be published within
> the nearest future, deals with some ghost like
> phenomena I've found when playing with this formula.
>
> --------------------------
>
> Regards,
>
> Ingvar
>
> www.come.to/kullberg
>
>
> LovelyBrot {
> fractal:
> title="LovelyBrot" width=640 height=480 layers=1
> credits="Ingvar Kullberg;9/1/2006"
> layer:
> method=multipass caption="Background" opacity=100
> mapping:
> center=1.149972209735855/0.273915609240573 magn=341141.33
> formula:
> maxiter=50000 filename="ik3.ufm" entry="ExtendedCompasses"
> p_PlottedPlane="1.(a-real,a-imag)" p_hide=yes p_areal=0.0
> p_aimag=0.0 p_breal=0.0 p_bimag=0.0 p_xrot=0.0 p_yrot=0.0
> p_xrott=0.0 p_yrott=0.0 p_zrot=0.0 p_exponent=-4.54545354/1.875778
> p_LocalRot=no p_diff=no p_bailout=100000000 p_dbailout=1E-6
> inside:
> transfer=none
> outside:
> density=2 transfer=linear
> gradient:
> smooth=yes index=0 color=8716288 index=100 color=16121855 index=200
> color=46591 index=300 color=156
> opacity:
> smooth=no index=0 opacity=255
> }
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