I’ve already asked folks in some nearby math departments. So far all I get is shrugs, so I’m trying here:

By plotting attractors (and near attractors), I found structures inside the Mandelbrot set, by which I mean, inside the parts that are usually black (or some other solid color). There are some photos here:

And a description here:

So far, I have yet to find evidence that any real mathematicians know this stuff exists. But it’s hard to believe I’ve stumbled across a new thing. Can anybody here point me to a source that explains what’s going on?

I’m not sure if the self-similarity you refer to is involved, but yes, I gather each genuine point in the Mandelbrot set is an attractor. My plots are based more or less on how quickly each point’s series converges. I assume the twisty threads in the plot indicate sets of points whose series converge the same way.

Given that holograms are just fringes too I have wondered for a decade or three about the following. Produce a Mandlebrot (or other fractal) “fringe” image (pick your favorite area). Take a high resolution image of that. Then shine a laser on that negative and see what kind of 3D hologram THAT produces. Alas by the time I get around to doing that the film technology required to do so will probably no longer exist.

Well, yes, the fractal features show up at the border of the set. Practically speaking, however, the classical Mandelbrot plot only colors points that are external to the set. And the “new” features I’m asking about are entirely within the set (points that previously have always been colored black or some solid color).

If it gets confusing, you can press X in my applet to exclude exterior or interior features from the plot. (Each press of X excludes exterior, interior, or neither in rotation.)

Each keypress starts a new calculation, so it can get slow. Sorry.

I don’t think the word “fringe” means the same thing with respect to holograms. A hologram is a photographic record of light wave interference patterns. While Mandelbrot images might involve mathematical interference patterns of a sort (particularly the internal ones I’m exploring), it seems unlikely that you could read them meaningfully as a hologram. Perhaps you could apply that idea to a useful computational analysis, but that’s way out of my depth.

Can you give us the coordinates of any of these features? I have a Mandelbrot program lying around that I could double-check them with. Or, of course, someone could just go through by hand and see where that point leads.

It sorta does. You look at a film hologram at high resolution and its a series of “fringes”. Obviously in black and white and analog rather that digital so you would probably want a gray scale fractal rather than binary. A color fractal would only make sense if you were doing a color hologram with color film and mutli colored laser light to “replay” it (lets crawl before we walk ) So, if you take a fractal and shrink it down to “about” the right size, take a high resolution photo of it, you should get a hologram of “something”. Might be a boring cloudy sorta thing. It might be a cool assed 3D series of intertwining filaments. You probably couldnt pick any old region and get something interesting. You’d have to be a bit smart about it I suspect. Most fractal areas might not generate anything. But some might.

For that matter, you can computationally take a fractal fringe pattern and calculate what it would create “holographically”. But there would be two problems. First, you could only calculate what it would look like at one point (okay two if you want to do a stereogram). Second, it would be extremely computationally intensive. Like many orders of magnitude more intensive than it would be generate the fractal in the first place. It wouldn’t be easy I know that. Now whether its in the realm of reasonably doable I don’t know off the top of my head.

Most or all of the images have the origin and width printed at the bottom. Open the image and zoom it to full size. The displayed coordinate resolution is kind of pathetic.

My program doesn’t have any facility for entering coordinates, sorry. I doubt any other Mandelbrot program will display the internal features. That’s sort of the point of my question. If it does, please let me know!

With my applet, you can find similar features pretty much anywhere just inside the set’s border. Just increase the sensitivity until stuff starts showing up.

Well, mine, I wrote myself, and I didn’t include any cleverness in auto-filling the black parts. So if they’re there to be found, I should be able to find them.

I’m genuinely confused here. It’s been a long time since I wrote a Mandelbrot program, but as I recall the points inside the Mandelbrot set don’t converge.

The iteration calculations inside the set do tend to get stuck in iterative loops, though (where eventually Z(m+n) = Z(m) ). Mark Peterson used that trick to greatly speed up the FRACTINT program when it was calculating points that turned out to be in the set. (After you’ve run a certain number of iterations start keeping track of the results of the last N iterations and if you find that the result of your current iteration is the same as one of the previous ones then you’re in a loop and the point is inside the set. Periodically increase the value if N as your iteration count increases. )

So far none of the stuff I’m seeing inside the Mandelbrot set using your applet has the rich intricate detail of the exterior. I’ve tried zooming in all over the place using your app and toggling X and the stuff on the outside is always more interesting to examine.

The fact that features change dramatically or disappear with one increment or decrement in the number of iterations makes me think that they are caused by rounding artifacts in the algorithm you are using. The exterior features only get more or less detailed when changing the number of iterations, they don’t jump around.

Apparently I was the one who was confused there, as I thought you were using the word “converged” to mean that eventually Z(m+1) = Z(m). I tend to think of a series that eventually settles down to a pattern where Z(m+n) = Z(m) where n > 1 as a series that gets into an infinite loop rather than one that converges. Blame my programmer mentality.

I dimly remember someone modifying FRACTINT so that it displayed points inside the set in different colors based on the length of the iterative loop (“n” in the above paragraph) rather than just in black, but I don’t know if that modification ever made it to a public release.

And I don’t recall anyone thinking of it as important mathematically - it was just another way of generating images. That doesn’t mean there wasn’t something important involved mathematically, of course.

I don’t claim the interior features are prettier. My point is they’re a hell of a lot weirder than the black blob that’s usually presented, and some of the patterns are downright freakish.

I also agree that rounding is significant. In fact, it is by aggressive rounding (the “sensitivity” setting) that I made the twisty patterns visible. Without rounding, you have to drill down deep into the set, and the patterns are only visible as dot patterns that gradually resolve to lines after you zoom in far enough. Unrounded, these lines are usually visible only as single traces, except where one or more of them intersect. Which is cussed hard to find.

Small iteration changes can vastly change the pattern. For example, compare these two results, which differ only by one iteration:

I suspect changes based on iterations occur because the algorithm I use most employs the count-out value to test for repetition, rather than actively seeking the first value that repeats. I suspect the math involves a periodicity that can make this algorithm vastly more or less sensitive based on the final count.

Rounding and algorithm quibbles aside, I don’t believe those factors alone actually create the patterns I’m finding. Even if they do, it represents a mathematical relationship that is far beyond my ability to explain. I’d still like to know if someone, somewhere, with the necessary mathematical chops, has some idea what’s going on.