FOTD -- January 04, 2003 (Rating 5)
Fractal visionaries and
enthusiasts:
For quite some time I have been hearing rumors of the existence of fractal things known as Siegel Disks. When I first started hearing these rumors, I had no idea what Siegel Disks were supposed to be. I supposed they were some type of
Julia set, but why were they called disks? I wondered about the problem a day or two, then forgot it.
Then, a few months later, I found that a Siegel Disk is a feature that appears in
Julia sets with a C-value that lies on the exact boundary of the
Mandelbrot set. And there appears to be a particular
Julia set, the one with a C-value of -0.390541,-0.586788i, that lies on the boundary and is considered to be the 'Siegel Fractal'.
At that point, I lost interest in Siegel Disks, but yesterday, I decided that I would like to see exactly what a Siegel disk looks like, so I calculated the
Julia set that corresponds to the above value of C. The result was a
Julia set like many I had seen and ignored before. Why, I wondered, was this fractal called a 'disk'. I raised the
maxiter from my usual exploration value of 1200 to 500,000 and recalculated the fractal. Other than going somewhat slower, the image remained unchanged. But I did not give up.
Doing further investigation, I changed the value of C to -0.390547-0.586796i, a value that corresponds to a point lying a tiny bit inland on the
M-set, and recalculated the fractal. To be sure the calculation did not skip over any near-periodic points, I set the
periodicity to 12.
Now, things started to happen. As the image developed, I saw that the fractal was not empty, but was filled with disk-like rings of extreme maxiter value. Here at last were my Siegel Disks. They are not the true Siegel Disks, which would have a maxiter of infinity and be totally flat and featureless. These would be impossible to draw. My disks are close enough to be called pseudo-siegel-disks however, and they have a huge advantage in that they are possible to calculate within the time that the universe will continue to exist.
When all my exploration was finished, I named the resulting image
"Siegel Disk Fractal" and rated it at a 5. The rating is mostly for the mathematical interest of the image. Artistically, it rates no more than a 3 or 4.
Despite its extreme maxiter, the image draws in only 15 minutes on my tired old
Pentium 200mhz machine. Most machines, being faster, will draw it faster. And as always, the completed GIF image is available on the internet at:
and at:
Now that I have a way of controlling the flood of
spam that could result, I also might soon resume posting the FOTD images to:
But don't go there yet. I have not yet made the decision.
It rained and rained Friday here at
F.C. -- a 34F 1C ice-cold rain that did good to neither man nor cats. The man passed the time accomplishing some useful work. The cats passed the time sulking like the mature cats they are. Whatever became of the happy, playful kittens they were 10 years ago?
Today is starting cloudy, but a clearing is on the horizon, and
I expect the day will turn out far better. And I have only a modicum of work to finish, which should make the day a good one by the time it ends. Until the next FOTD appears in 24 hours, take care, and I wonder how much data a Siegel disk can hold.
Jim Muth
jamth@mindspring.com
jimmuth@aol.com
times.
