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All the images were created using xfractint 19.20, the unix port of fractint. Fractint supports encoding the image parameters into a gif file but the unix port has a bug. As far as i can see this bug causes only the calculation depth parameter to be corrupted. After loading it usually is too low so recalculation gives you only a black image. Just set it to a reasonable high value and have fun.
All images are copyrighted by Robert Figura. Copying and linking to is allowed for non commercial purposes as long as my name and copyright are mentioned nearby.
Early in my childhood i saw my first
mandelbrot fractal which immediately fascinated me. This fascination
kept plagueing me and my environment and probably will until i
die. Some fractal images are beautiful and some are interesting but i
was more interested in the general structure and not searching for
just a single nice spiral. The mandelbrot fractal contains an infinite
set of spirals and given any properties it should be possible to find
optically appealing instances for some given properties. So i started
to explore the structure of the mandelbrot fractal in a half random
and half systematical way. I restricted my search to mandelbrot sets
since julias are just boring and other formulas cause superflucios
complexity.
[ What? You never heard about mandelbrot fractals? In this case you
probably want to read this as an
introduction to fractals. It contains a description on how an
image of the mandelbrot can be generated. ]
I started to form certain notions to better remember the way i took in descending the abyss of structure. The main motive i call (how cold i do else) mandelbrot. See the image on the right side for the main notions. Further i call a child wrapped in symmetric structure Temple. In my very humble opinion the word spiral does not need further explanation. Lake i call any continous area of color, the lakes that are part of the mandelbrot set are called internal lake. After i created these notions, i did my first systematical descent.
Every child knows what a spiral is. The mathematician knows about logarithmic, archimedian and other types of spirals. But mandelbrot spirals are special.
Lets first view the two essentially different types of spirals.
I found out that the dominant structures on the left hand of the upper
canyon contain thin spirals and on the right hand some kind
of inverted spirals can be found.
The normal spiral is a rather simple object whereas the inverted one
is a very complicated object. The inverted spiral can be arranged to
show another spiral structure.
[ I am speaking of it like of a creative act since all variations of spirals can be found inside the mandelbrot set. For me it feels mere like i am designing an image than i am finding one. After some years of practice you will doubltlessly attain the ability to tune the motives you want to show. ]
But i have another spiralous image which somehow looks like an
inverted spiral but it seems to have a strange symmetry as if multiple
spirals were combined. It reminds me of quasi crystals and of penrose
tilings. Unluckily the parameters were lost so you will have to find
out yourselves where in the mandel to find such oddities. Hint: I used
the logarithmic palette.
There are spirals with more than one arm wrapping around it. Take for
example the image on the right with a double spiral in it. Follow one
arm for one round. See? At each family member deeper into a canyon the
number of branches increases one by one but the arms of the spirals do
not increase. A way exists to increase the number auf arms
exponentially. Here are the instructions: Zoom into a double spiral;
select one member and zoom to the place where the biggest children
is. You may have to try it out on a rather simple spiral first. There
you will see a quadruple finite spiral with a child in the center. The
deeper you zoom into the first spiral, the more often it is wrapped
around its child. Where the quadruple breaks up, an 8-spiral waits.
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Now the basic Spiral knowledge ends but haven't i told something about
artwork? Here is a spiral composition which i consider beautiful. It
is not only a spiral but also a temple so this picture is perfectly suited to lead to the next section:
A temple is a symmetric structure with a mandelbrot child in its center. Most of the fractals images i created have a temple in its center and therefor i say that an image shows a temple if its four similar structures can be seen as the dominant element of the picture, as most buildings have 4 main walls. We have already seen a spiral temple, so note the following instructions:
Zoom onto the biggest child on the antenna; enlarge the fine line deep in
the back of it; There you will find a very symmetric temple. The image
of the big bang on the left is a slightly perturbed form of such a
temple i selected from another antenna.
I used to have a variant of such a temple as background image for my
desktop. This one again is a little riddle since i did some image
processing to emphasize the child in the center.
[ It looks like i have to fiddle around by hand so my browser manages to display these floating images without producing a mess for my two favorite sizes of the window. But if i am always talking that much about it then i will have enough text to insert and my browser will do fine work... It looks like i am not able to tell that much about fractals as i expected i would. Anyway this maybe is the right place to apologize if your configration does not allow the proper visualization of this website. ]
Complex inverse siprals give a very good basis for beautiful temple
compositions. If you'd like to take your eyes to the left you may grab
a look onto a 4-fold structure which would make a nice motive for a
wristwatch for days with 4*23 hours. Apropos 23.
The two triangular structures on the right do resemble a temple and
have a mandel inside. I especially like the brown parallelogram around
it. Also note that you can also see an other kind of spiral gathering
where two simple spirals stick around a two armed central spiral.
Some of my spiralous collection can be found in the table left below. Note also the sun motive which can be located easily at the far left and which is an example of a carefully adjusted color scheme. Ummm where was i? Ah yes. Temples. I told already that i have lots of them? They're multiplying like plant louses. This is no surprise since any of them has a child inside. Well some are more and others are less pregnant. Interesting structures can be revealed if you zoom in a zig zag fashion on alternating branches.
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These examples show how profoundly different shapes can be found among the simpler objects inside a mandelbrot by systematical approximation.
[ I seldom use fractint's deep zooming facility for it is too slow for my interactive approach. So nearly all of the images were calculated using the limited accuracy of standard floating point numbers ]
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Of course there are many more structures coming up again and again. I Would like to mention on more. A swirl is a short wave structure. Since i have not found a swirl without a child in its center these may be considered as a special kind of temples. Swirls are related to san pedro forms [ BTW: I stole that nifty san pedro notion from a book about fractals. ] in the way that a san pedro is a limiting case of a swirl. See that big sea-green image on the right for something in between.
Also both, swirls and san pedros may be found with in similar
places. Here the short instructions: Antenna - child - canyon -
antenna - san-pedro. I prefer bifurcation antennas.
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But also the more chaotic structures are interesting. The complicated temple on the right is very beautiful. Note also the freaky chaotic bubble cross. Many structures with these bubble like lakes can be found deep inside canyons or backsides.
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| Links to Software |
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| Fractint User Homepage A fractal generator software that is |
| XaoS homepage Software for realtime animated zoom into fractals |
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mailto: Robert Figura |