Fractal eXtreme: FXMovie - Really Deep Zoom

Fractal eXtreme Zoom Movie - Really Deep Zoom

This image is from an area that was first discovered using FractInt. I believe the original explorer of this area was Dewey Odhner. It looks rather unfractal like. In fact, the horizontal stripes look like something that could not possibly be found in the Mandelbrot set. This caused some consternation when this was first found.

The reason for the concern can be seen even more clearly when you change the maximum iterations per pixel to 2305, thus showing nothing more than a ring of spokes - not quite the chaotic images we expect from the most complex object every found.
These two pictures below look even less fractal like. Here the number of spokes has gone up even higher. There are 2048 of the red spokes, and we can see in the right hand image the point where the bifurcation occurs and 4096 green spokes are born.

You can also just barely tell that all 4096 green spokes are, in fact, one connected solid region. In fact, the red spokes are all at an iteration count of 2816, and the green spokes are entirely at an iteration count of 3072.

By making use of the zoom movie capability of Fractal eXtreme we can make it graphically clear how these "impossible" images occur.

For best results, if you have a true-colours graphics card and a P166 or better, you may want to make sure that you have the latest 1.11 version of the Zoom Movie player, because the optional bilinear scaling introduced in that version allows many annoying aliasing artifacts to be removed.

A zoom movie showing the complete path from the unzoomed Mandelbrot set to this area would be about five minutes long - and most of the time there would be nothing of interest happening. I've captured some of the highlights in the partial zoom movies, and annotated all of the highlights here.

Throughout this discussion, bear in mind that a 'zoom' is a doubling of magnification, and approximately one hundred and forty zooms are all that are required to magnify an electron to the size of the universe.

The zoom movie starts by zooming in towards the point (-2, 0). This is the left edge of the spike on the left side of the Mandelbrot set - also known as "utter west."

507 zooms - finally we 'break away' from utter west. For the first time we can see that we aren't zooming precisely towards -2, 0. Go back to sleep, because nothing of interest happens until...
764 zooms - our zooming image zooms in on two new spikes - now we have 4.
891 zooms - number of spikes doubles again, to 8.
955 zooms - number of spikes doubles again, to 16.
987 zooms - number of spikes doubles again, to 32.
1003 zooms - number of spikes doubles again, to 64.
1011 zooms - number of spikes doubles again, to 128.
1015 zooms - number of spikes doubles again, to 256.
1017 zooms - number of spikes doubles again, to 512.
1018 zooms - number of spikes doubles again, to 1024.
1019 zooms - our destination minibrot is the size of our screen. The number of spikes continues doubling, but very high resolution is needed to resolve them. Fade to black.

The zoom movie to the right joins the action at 977 zooms, when there are only 16 spikes, and continues zooming until there are thousands. As you can see the number of radial bands, or spokes, keeps on doubling as you zoom in until there are dozens of spokes visible running almost parallel to each other. In this area you can find dense groupings of spokes going in any direction desired. It took a PII at 300Mhz approximately seventeen hours to calculate this movie. While this seems like a long time, it's actually very fast considering the length of the movie and considering that the calculations are being done to over three hundred digits of accuracy! And considering that the machine was being used as a full time software development machine while the movie was rendering. that's pretty darned quick!

The coordinates of the center of the image are approximately:

Z Real = -1.999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 999, 998, 895, 836, 912, 417, 836, 217, 316, 527, 999, 933, 164, 430, 805, 531, 612, 097, 027, 084, 085, 942, 478, 631, 470, 944, 179, 784, 765, 553, 302, 325, 289, 533, 320, 961, 898, 109, 921, 686, 419, 831, 529, 734, 837, 424, 289, 096, 210, 971, 806, 586, 916, 814

Z Imaginary = +0.000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000, 000

Zooms = 1019

This image of a mini-brot at much lower magnification shows the structures that our circle of spokes evolve from. Although the magnification of this image is just three thousand times, or a little over eleven zooms, we can clearly see the pattern of spikes where each new set of spikes is shorter, but twice as numerous as the set before. As you find mini-brots which are at higher magnifications, the spikes get straighter, and the difference between their lengths gets greater.

Each spoke is made up of a single iteration band. Oddly enough, there is a very simple relationship between the iteration count of that band and the number of spokes, for the part of the Mandelbrot set in the movies above. The formula is:

	NumBands = 2^(NumIters/256)

or, alternately:

	NumIters = lg2(NumBands) * 256

To put this in slightly less mathematical terms, every time the number of spokes doubles, the number of iterations for the next set of spokes goes up by 256. I suppose it is inevitable that there be some relationship between the two, but I hadn't been expecting anything quite so simple!

Now seems like a good time to remind everyone that there is precisely one iteration band for each number of iterations, and iteration bands never cross. Therefore, when we have 4096 spokes, they are all connected at their outermost extents. The two spokes that are closest to horizontal have the entire Mandelbrot set in-between them, which is a pretty hefty obstacle to go around when it's been magnified by one thousand or more zooms!

Also note that the binary appearance of the spokes is presumably related to binary decomposition of the Mandelbrot set and to the calculation of external angles.

I need zoom movie plug-in help!