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The Real Mandelbrot Competition

This competition has very strict rules:

1) Only images of the Mandelbrot Set are allowed.
2) The only colouring method allowed is the traditional banded method: iterations are mapped to colours.
3) The only post processing allowed is anti-aliasing/sub sampling.
4) Images must be verifiable as part of the Mandelbrot set. To this end, precise locations must be provided with each image.

What the judges should look for:

1) Creative zooming: unusual shapes and forms created by the effects of zooming into the M-set.
2) Images which suggest the image is not a part of the M-set
3) Images which are at very deep depths within the M-set which require arbitrary maths precision.

What does this mean?

It means that beauty is not the sole criteria for this competition. Ugly images are welcome and encourage! It means that beauty is welcome and encouraged, but must also fulfil knowledge of the M-set creatively applied. It means that the images will probably take a long time to render.

Because of these considerations, image sizes may be as small as 400 x 400 pixels, or as large as you feel sensible.

Prizes
My respect!

Any takers?

[edit]
This is something I'd genuinely like to see. I've seen the deep zoom animations on youtube and elsewhere, I've read about how long these things took to render.  I'm curious as to how the final coordinates were chosen? Was it an algorithm that chose them, or did you zoom in with an image generator until you found the coordinates and then made a video zooming to those points? If the latter, which is what I'm hoping, surely you discovered some interesting images along the way? What insights did you glean? Use them!

I admit, that this past week is the first time I've been able to get past the 64bit precision barrier of double precision maths, and so it's still exciting and I'm not sure what to expect. Will what I find live up to my expectations!?!?

I think this is a good idea and I'd gladly enter.  One other thing I'd like to see is a short blurb about how/why this particular image was chosen.  If it's just because the contributor likes it, that's fine.  But if there was some definite process/algorithm for finding this zoom, particularly for deep zooms, it would be good if that were shared.

Ah good :) I agree with the blurb about choices made, and the sharing of the algorithm. Although a part of me wants to make it a rule that no algorithms other than wet-ware based algorithms may be used. But maybe that's taking the restrictions too far? I do want to see images that have been thought about, though equally I don't want to discount more impulsive zooms as they can often lead to new insights.

I feel I should justify the tight rules a little. Artists have always used restrictions and limits in what the allow themselves to do, it's as an important aspect in art as is freedom. Particular examples escape me, but practices such as colour palette restriction, the novel where none of the words contained the letter E, etc.

There is one thing you might want to clarify, for fairness -- does it matter how the iterations are mapped to colors?  I would suggest, at the least, that you allow a logarithmic mapping.  The reason for this is that generally, the more you zoom in, the smaller the color bands get.  Once you get fairly deep, they really don't even look like color bands any more, just random static.  The bands essentially become much smaller than a single pixel in width, so what you see just looks like noise.  But if you use the logarithm of the number of iterations, you get back to nice, smooth color bands.

Many people also prefer the smooth gradients (as opposed to distinct bands) that are produced by "normalizing" the iteration count.  The color bands are really just artifacts of the bailout algorithm used.  Here are images illustrating the difference (from Wikipedia):

Not normalized:
(http://upload.wikimedia.org/wikipedia/en/9/9c/Escape_Time_Algorithm.png)

Normalized:
(http://upload.wikimedia.org/wikipedia/en/0/01/Normalized_Iteration_Count_Algorithm.png)

The colour mapping of iterations maybe scaled such that a 256 colour palette may be stretched across several thousand iterations with the in-between colours formed by interpolation.

The colour bands may not be replaced by a smooth gradient. This defeats the purpose of exploration and deep zooming. If you want a smooth gradient you only need zoom deep enough. Here is an example:

(http://www.fractalforums.com/gallery/1/thumb_1095_06_01_10_1_34_31.png)
http://www.fractalforums.com/gallery/?sa=view;id=1274 (http://www.fractalforums.com/gallery/?sa=view;id=1274)

I personally find this more interesting than a purely smooth gradient. In this image, iterations were multiplied by 0.01804680000000000173 to get the colour palette index.

I too like this idea!!!

But are there limitations as to the rotation of the slice being presented??  And, by post-processing, does that also mean no "layers" being merged together, just a single slice of the M-Set??

And yes, Kerry's suggestion is a good one.

No layering allowed either! No rotation!

Hang on.... I'm starting to see a pattern here... All the things that are not allowed are all things the program I've written can't do! But seriously, my basic idea is...

Another thing, an alternative title was "The Imaginary Mandelbrot Competition", or if I'm feeling really indecisive, "The Real Imaginary Mandelbrot Competition".

My basic idea I guess evolved from... The limitations of my programming and maths skills... Well that's not quite true, I've implemented different fractal types, colouring methods, and auto-layering, but... I have this crackpot idea that if I only zoom in deep enough, and make the right choices, there is something in there, waiting to be discovered! So having all these strict rules (I'm mainly trying to work this out for myself here) about what cannot be presented is to place the emphasis on the (creative) possibilities in the set itself, on exploration, on insights, etc, etc.

Well, the normalized color gradients are actually more "correct," if there is such a thing.  The bands are just artifacts (computation errors, in other words) that show up because of the choice made to "bail out" at some hard limit.  People have become accustomed to them because pretty much 100% of the old fractal programs had them.  The Mandelbrot set itself doesn't have distinct bands.  But whatever, if you prefer more of a "retro" contest, that's cool!   ;D

Oh I see! Hmm. Maybe if I can find information on how that smooth colour gradients are rendered and can implement it, the rules might change ;-)

I was just looking in the programming section at http://www.fractalforums.com/programming/antialiasing-fractals-how-best-to-do-it/ (http://www.fractalforums.com/programming/antialiasing-fractals-how-best-to-do-it/) which lead me to HPDZ.NET (http://HPDZ.NET) and the smooth gradients don't look that noticeably different at that level.

Getting OT here, if I do attempt to implement the normalized smooth colour gradient, I hope/expect that it should be able to be stretched out in the same way as I was talking about with the colour bands.

Furtherly OT, I'm still working on the multi-threaded version of mdz Mandelbrot Deep Zoom, it's working but anti-aliasing is borked, and just to get it working I did not bother using the colour palette and it just converts the iterations/colour bands to b&w values. Just got to try and figure out how to put it all back together again.

Here's another image to get things started:

(http://www.fractalforums.com/gallery/1/1095_13_01_10_9_47_22.png)

( A screen shot because image saving is mysteriously broken in my multi-threaded experiment with mdz, as is anti-aliasing (or more accurately super-sampling) )

When i saw the name i thought it would be a competition for images computed using real numbers only.... now that would be a good challenge.

Huh, I probably sound like complete noob saying this, but I had no idea things like this lurked in the M-set. I really don't have any knack for good fractal exploring.  :(

It takes a bit of time to get really good at fractal exploration, learning the many areas and various parameter settings.  And some people have more luck on their side than others.  Either way, it is still a very time-consuming interest/hobby.    ;)   :D

I have spent many hours, days, months, years, and still have much to learn and explore.

Here's one.  I've included the Ultra Fractal parameters below.  For those who don't speak UF, here are the center point and magnification:
  x = -1.74734877811384949048239826095235961267258435541015
  y = 0.0022865486943484368266400015529741555193263742956452
  magnification = 4.1e39

I found it by starting at the west midget (largest midget on the spike, center at about -1.75).  I zoomed in to the 1/10 disk and found an embedded Julia on the filament leading to the tip of the disk's structure.  Then, I zoomed in several times, alternating between concentrating on tips and centers of embedded Julia sets.  What I like about this image is that it illustrates the irregular path taken by orbits of points taken from this general area.  Normally, we may be used to thinking about period doubling as a route to chaos.  That's reflected in a doubling of the order of the structure surrounding a midget:  2-fold symmetry surrounding 4-fold, surrounding 8-fold, etc., until it all collapses into visual noise around the midget.  Here, we have an 8-fold structure surrounding a 4-fold.  Going into the center of the image, it proceeds:  8, 2, and 4 and probably more back-and-forth before getting into the final period doubling that leads to the midget.

jan15-a {
fractal:
  title="jan15-a" width=3000 height=3000 layers=1
  credits="Kerry Mitchell;1/15/2010"
layer:
  caption="Background" opacity=100
mapping:
  center=-1.74734877811384949048239826095235961267258435541015/0.00228\
  65486943484368266400015529741555193263742956452 magn=4.1e39
formula:
  maxiter=10000 percheck=off filename="Standard.ufm"
  entry="Mandelbrot" p_start=0/0 p_power=2/0 p_bailout=4
inside:
  transfer=none solid=4294901760
outside:
  transfer=linear solid=4294901760
gradient:
  smooth=yes rotation=125 index=125 color=0 index=325 color=16777215
opacity:
  smooth=no index=0 opacity=255
}

"The Reason Why We Always See Bums in the Mandelbrot Set" (don't ask)

(http://www.fractalforums.com/gallery/1/1095_18_01_10_11_42_38.png)

xmin -1.7491976289657893741942376816272921165326158557715556129946
xmax -1.7491976289657893741942376816272921165326158557113309627591
ymax -4.2530777152440422725855012159249401150953497611785388221839e-7

I've just found an article from The College Mathematics Journal, Vol. 26, No. 2, (1995), pp. 90-99 which presents a bit of a challenge to this "Real Mandelbrot" competition idea. To quote from the conclusion:


The article is titled "Can We See the Mandelbrot Set" downloadable PDF from http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2690 (http://mathdl.maa.org/mathDL/22/?pa=content&sa=viewDocument&nodeId=2690)

I don't understand the mathematics of his argument, but I'm sceptical that images of the Mandelbrot Set are wrong as such. Part of his argument seems to be that the pictures are misleading because they don't show the M-Set is connected...

On the other hand, it was some time before I understood that images of the M-Set are representations of it, not the actual M-Set.

I found this article very weird. Of course no one can really see the M-set, but I think all the representations done for many years do have a strong enough mathematical background to be able to say they are very close to the real thing. I don't understand why one has to be skeptic about that.

I ignore such comments on rendering fractals since:

1. It's patently obvious we can only see it *to a given resolution*.
2. For a given render if the accuracy (float,double,arbitrary), iterations and bailout are high enough then increasing the accuracy further does not alter the image at all.

This competition is really captivating

This is not really a competition entry - I don't properly maintain revisions of my own multiprecision Mandelbrot explorer program, so I probably lost the exact location information. But if you want "unusual" structures that are never seen in the usual crop of Mandelbrot renderings, here is one in three different instances:

http://www.vectorizer.org/mandelbrot/mandelbrot.html

Very deep zooms can indeed reveal new structures that do not exist at lower magnification.

It's not a formal competition as such, but anyway...

I like your three images. At first I thought, huh, I've seen those 4-pronged stars enough already, but looking at the higher-resolution images revealed a few nice surprises, I really like them - the selective zooming is evident in the shapes that form the stars, and I can recognize their genus, but am unsure if (particularly CloudPillow.png) I can quite recognize the traces of deeper elements if you know what I mean?

Thanks,

James.

BTW, I've given the fractals a bit of a break for now, but I shall return at some point of course! Sometimes though, like many other creative processes, having a break from zooming into the M-set can be a good thing, especially if you keep on returning to the same-old ideas of where/how to zoom into it.

Yes, I think I know what you mean. The things you call "deeper elements", I purposefully overwhelmed them by making the star larger and larger. In a way, you can choose some properties of these stars arbitrarily.


Let's see if I can explain how they were "built" ... it all starts from an older method of layering two distinct features of the Mandelbrot set into a single image. Basically, if you have two images that you want to mix, do the following:


1. in each image, identify the relative position of the desired structure with respect to the main cardioid.

2. find a (not too small, not too distorted) satellite cardioid, i.e. a minibrot in each image

now you have two opportunities to combine the shapes, depending on which of the two original images you start from (I suggest trying both):

3. in image A, zoom into a minibrot. In that minibrot, close in onto the relative position of the features of image B. Sometimes this is already enough to obtain an image with both structures. But at other times, you need to

4. zoom towards one of the embedded julia sets, until finally an image with combined structures emerges


That's the basic building step for "constructing" Mandelbrot images. One structure is a "primary" feature, while the other one will have a more ornamental "secondary" role. You can control which is which by choosing to start from either image A or image B.

As you can guess, the described two or three phases of deep zooming add up to unusual magnification factors. This is why arbitrary precision is needed if you want to layer many structures on top of each other.



Now, on to the stars. I found these by layering a particular structure onto itself, time and time again. Here's a general guide:

1. find any spiral in the image with your desired structure (if all else fails, there will be spirals near minibrots)

2. find a minibrot in this spiral, maybe one or two turns into the spiral (too many turns do more harm than good)

3. that minibrot will have "ears", i.e. two smaller spirals on opposite sides.

4. zoom into one of the ears, maybe one or two turns into the spiral

5. where you found the minibrot earlier, there will now be an embedded julia set with the familiar two ears

6. zoom into the center of the embedded julia set until you find that the ears get another ear each

... so we just layered more ears on the ears, and can do so as many times as we like.

Repeat this for a while, and gradually the string of ears becomes so long that the emerging meta shape, a simple straight line of "finite" length begins to dominate. Then find the centerpoint of the line, and zoom into the embedded julia/minibrot until the phase has doubled. This finally turns the line segment into a four pronged star.


Other nice shapes to layer onto themselves are ornamental trees, like those around the largest minibrot in the long spike. But with such intricate plane covering structures, you need a distance estimator and render just the borderline, otherwise the images will be overloaded and noisy.

Still, it would be quite challenging problem to render the set in such a way that any detail would be visible no matter how small they are. Normally, anything that adds less than 1/256 of particular color range to the image blends with surrounding pixels and so is effectively invisible.

I can't imagine it being all that difficult to render infinite detail in finite space either  :o

I agree - I think he totally left out everything about how iteration coloring can clearly show us where the Mandelbrot set is. He never even mentioned anti-aliasing. We don't need to see it colored black to know that there's something there. And he kept on talking about "round-off errors" and "inappropriate thresholds": We have arbitrary precision, and for the second (depending on which threshold he was referring to) we have millions of iterations and, again, colors to show us the fine filaments.

if that's easy, how about someone actually make an image that does show it "colored black"?

If anyone doubts the accuracy of a deep zoom, then render it again with 10* the anti-aliasing and max iterations, if it hardly changes then quit worrying.
You hardly see the fishing line between hook and rod in a picture. It doesn't make the picture wrong, just means the line is thin.

and it also means you can't see it in the picture. which was kinda the point, wasn't it.

Not really. If you try and draw cantor dust using a black pixel if anywhere under the pixel is in the set.. then you get just black. If you render a spider web from a distance doing the same thing you get just black. Anti-aliasing is a better representation I think. It render the _amount_ of points that are in the set under that pixel.
It is of course an approximation, but as the anti-alias resolution goes up, the correctness of the pixel approaches 100%.

this would be true if you had an infinite range of colors, which you dont. also, even if you had, human eye also has its litmits, so you wouldn't actually see very slightly gray pixel different from surrounding white (assuming AA-ed B&W set picture).

that's not to say that painting every pixel with non-0 intersection with set black is better method, but it could allow us to see it where we currently cant.

I was recently looking at actual Mandelbrot orbits (http://wonderfl.net/code/9845741d6fac7dfe266e874e95a1917df00bdfba) and found that, while many orbits clearly diverge or converge, some vibrate for a long time, and some of those some suddenly diverge or converge after significant number of iterations. Does anyone have any idea if this is caused by limited floating point precision, or is it real behavior?

Generally speaking it's real behavior - this can be checked for quite simply - if the behavior changes as the precision used is changed then it's a limitation of the precision used, otherwise it's "real".

on the other hand, assuming orbits that start outside the set diverge, and those which start inside converge, what is expected behavior of orbits that start exactly on border? I assume these are stable orbits that remain the same over time, and I see a lot of these (at leas visually so, who knows what would happen if I wait few years). this line of thinking doesnt seem to have any room for orbits that initially converge and then suddenly diverge after long period of time... hence I was thinking it's some kind of accumulating error.

I've found that points very close to the boundary but actually "outside" to exhibit the behavior you describe - at whatever precision - basically the path to infinity is always an exponential one after the final spiral :)

a point really exactly on the edge would be -2, as far as I know. It's on the very outest edge of convergence. But it converges just as usual. To 2 :)
-2 -> 2 -> 2 -> 2

Points on the boundary generally have unstable orbits.  Consider c = -2 and the standard Mandelbrot set.  The exact sequence is 0, -2, 2, 2, 2, etc.  Computers won't have any difficulty with this since all the values are integers, but if that weren't the case and z got tweaked to 2.000001, the orbit would diverge.  Or if z got tweaked to 1.999999, then the orbit would (probably) be chaotic.  Or, c = 0.25; the orbit converges to z = 0.5.  If z is tweaked to something larger, then the orbit would diverge.  Tips of dendrites are unstable Misiurewicz points (like -2 or (0,1)) and points that are the tangent points between two disks have unstable orbits that straddle the boundary between the periodicity of the parent disk and that of the child disk.

excuse me, there is no edge :D, a point is either inside or outside  :police:

well, anything "more distant" than the edge of the circle with radius 2 in the origin does diverge for sure.
And -2 sits exactly on that edge.

You could find nearly never escaping points by getting closer and closer to the edge (that can't ever be found exactly, true), by putting one point to |z|=2 and one to |z|=1/4, using the same argument for both and then iterating, until it's sure, wether the points converge or diverge and reweight accordingly.
that way, you'd get to some kind of edge after some time :)

...and the one other case so often forgotten: strange attractors. A point the doesn't map into an unstable orbit (Misiurewicz points) and aren't parabolic (connecting two disks) exhibit even weirder behavior; if memory server, (-0.8,0.2) is one such point.

EDIT: Sorry, that should be (-0.8,0.15). A similar is (-1.2,0.15). Another, "different" kind of strange point is the Grossman-Whatshisname band-merging point. I think one of the banners is actually a picture of it. If you zoom towards the p-3 minibrot, then back towards the p-5 minibrot, then back to the p-7, etc. you'll find they converge (alternating) to one point. It's also not a Misiurwecz point, I think, nor is parabolic. The MSet zoo is infinite, it seems..

It's true that a point is either inside or outside, the inside is composed of two sets, the interior and the boundary.  An interior point is one such that a circle of positive radius can be constructed centered on the interior point such that all of the points inside the circle are inside the set.  A boundary point is one such that any circle centered on the point, no matter how small the radius, will contain both inside and outside points.

Here is a way to find some edge/boundary points:  Let theta be an angle in radians, whose value is a rational number (like 1.5 radians).  Then, r = (1 - cos(theta))/2, x = r cos(theta) + 0.25, and y = r sin(theta).  The point c = x+iy is on the boundary of the main cardioid of the Mandelbrot set, but since theta is not a fraction times 2pi, c is not a tangent point of a disk.

So, both .25 and -2 are boundary points, right?
Both of them do converge but shifting them just a tiny bit away from the center will make them diverge, so any circle around them will have both types of points.

However, those two are most likely the least interesting boundary points :) - What's about the outer most boundary on the imaginary values, for instance? (However, that was discussed in an other thread, I think...)