The diety hiding in the m-set
dan...@autodesk.com
I don't know if anyone has tried this before, but I thought of a new way
to look at the mandelbrot set. The process is simply this:
1) initialize the array of exit values to all zeros.
2) for each pixel:
iterate the usual mandelbrot function:
for each iteration:
find the pixel that the Z value lands on and increment
the exit value for that pixel.
This method shows you which points in the complex plane are attractors for
the m-set, or at least it shows you where the Z value spends it's time.
I've posted the resulting picture to alt.binaries.pictures.fractals as
"m-attr.gif". It's colored with the lowest values in red and proceeding
through yellow, orange, green, blue and with the highest values as violet.
I scaled the values into the color ramp using a simple form of histogram
equalization that I developed for coloring mandelbrot images.
I find the resulting image fascinating. It is shaped somewhat like the
m-set but the features do not really coincide with it. It consists of many
spheroid-like shapes covered with distorted grid lines all wrapped in
lace-like wispy features and other strange geometries.
I began to wonder what aspects of the m-set cause the various features.
It then occurred to me that perhaps some of the features were being
generated from points that originally start off in the m-set, and other
by points from outside the m-set and that the image was just a composite
of the two.
I then modified my program to generate those two images. I first created
an image of the ordinary m-set which I then used as a reference array which
the modified program used to iterate only those pixels that are, or are not
in the m-set.
The picture generated from the pixels in the m-set looks somewhat like
the first picture, but a bit simpler (as expected). It looks like a set
of various sized spheroid shapes exploding outward from a cloud of red
smoke. I posted this one as "bigbang.gif".
A totally unexpected thing happened when I generated the other picture (the
one from the pixels that are *not* in the m-set). This one looks very much
like some sort of seated Buddha. It looks just like one of those Indian
paintings complete with eyes, ears, headdress, arms and crossed legs. It
seems to be wearing lots of ornate jewelry and clothing. I'm not kidding.
If I were a religious person I would certainly take this as some sort of
sign. I posted this one under the name "ganesh.gif" because when I showed
it to an Indian coworker of mine, he instantly recognized it as the god
"Ganesh" which is the one with the head of an elephant.
The nice thing is that you don't need to take my word for it and retrieve
my posted image because you should be able to generate the same picture from
the above instructions. It's also interesting because it won't really be
possible to zoom in much on the features because the picture is the result
of the effects of all the points not in the m-set.
All three pictures were generated using a max-iteration value of 100,000.
They are also very interesting and quite different when using lower values.
I'd be happy to hear what other people think about this technique and my
images.
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Daniel Green ___________ ___________
dan...@autodesk.com | | | |
| _________ | _________
| | | | | |
| | _______ | | _______
| | | | | | | |
---| | _____ --| | _____
---| | D | --| | D |
---| G| --| G|
wall-eyed stereo --> ----- -----
Ken Shirriff
> for each iteration:
> find the pixel that the Z value lands on and increment
> the exit value for that pixel.
I think this is similar to Noel Giffin's "mandelcloud" type in (x)fractint.
Ken Shirriff shir...@sprite.Berkeley.EDU
dan...@autodesk.com
He replied to my posting and we've discussed the subject, and yes, it's the same
basic idea. I don't use Fractint, so I didn't know about that option.
The difference in our methods is that the mandelcloud option does not distinguish
between pixels that eventually excape to infinity, and those that don't. When
plotted together, the "diety" is obscured in the noise from the pixels that are
*in* the m-set. I only saw it when I excluded those pixels from my calculations.
You could do the same thing if for each pixel, you do the above incrementing and
then, if it turned out to be a pixel in the m-set, run through the same calculations
over again, but this time *decrement* each pixel that the Z value lands on.
In other words, subtract out the effect of the pixels *in* the m-set.
I don't know if that's an option that you could or would want to add to Fractint
because it doesn't lend itself to zooming, but the effect is really startling.
