Competition Idea

[jwm-art]

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:

So... the next time you see one of those gorgeous pictures of hte Mandelbrot set, with swirls and dots and dainty patterns, that claims to represent the fine detail of an amazingly complicated set, I hope you will admire the artistry... and question the mathematics. I hope you will be a skeptic.
I am.

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

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.

[bib]

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.

[David Makin]

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:

So... the next time you see one of those gorgeous pictures of hte Mandelbrot set, with swirls and dots and dainty patterns, that claims to represent the fine detail of an amazingly complicated set, I hope you will admire the artistry... and question the mathematics. I hope you will be a skeptic.
I am.

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

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 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.

[neetusharma]

This competition is really captivating

[hobold]

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.

[jwm-art]

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.

[hobold]

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.

[makc]

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.

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.

[jwm-art]

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.

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 

[Timeroot]

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 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.

[makc]

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

[Tglad]

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.

[makc]

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

[Tglad]

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%.

[makc]

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.