; Date: Sat, 04 Jan 2003 09:46:13 -0500 ; From: Jim Muth ; Subject: [Fractint] FOTD 04-01-03 (Siegel Disk Fractal [5]) ; Id: <1.5.4.16.20030104094852.29b74724@pop.mindspring.com> ; --------- ; 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 ; ; ; START PARAMETER FILE======================================= SiegelDisk_Fractal { ; time=0:15:25.16--SF5 on a P200 reset=2002 type=julia passes=1 center-mag=0/0/0.97\ 12963 params=-0.390547/-0.586796 float=y inside=255 maxiter=500000 logmap=yes symmetry=origin periodicity=12 colors=0000000000000000000000000000000000000000000\ 000000000000000000000000000300600AI0EC0I60K00O00S0\ 0W00_00c00g03k0Co0Ks0Uz6ezEozNzzWzzezzozzvzvzzkzz_\ zzNzzEzz3zv0zs0zo0zo0zb0zR0zE0z30k00_00K0080000000\ 000000Kz0No0NR0N30R00R00R00_00e00o00v00z0Cz0Rz0ez0\ vz0zz0zz0zz0zz0sz0ok0kb0eR6bIC_8IU0NR0UN0_I0eE0kC0\ s80z30k00b00R00I0080000C00K00I0CE0_C0z00z00z00z00z\ 00z00z00z08z0Kz0Wz0hv0vv0zz0zz0zz0zz0zz0zz0zz0zv0z\ o0vh0vb0sW0sR0oK0oE0k80k30o00s00s00v00v00z00z00z00\ z00v00o00h30b80WC0UI0NN0IU0C_06b03h00o00v00z00z30z\ 63z8RzCszEzzIzzKzzKzzKzzKkvKWoKKkEEhC8e83b30_00W00\ U00R00N00K00I00E00C0080080000000000000003608C30UI0\ kW0zU0zR0zR0zN0zN0zK3zK8zIEzEKzERzC_zCez8kz8sz6zz6\ zz0zz0zv0zk0zb0zU0zK0zC0z36z0_z0zzzzzzzzzzzzzzzzzz\ zzzoz0_zIRz_UzRUzIWz8Wz0_z0_z0kz0vz0zz0zz3zz6zsCzz\ Ezz8zz3zz0zz0zz0vz0kz0ez0 } ; END PARAMETER FILE========================================= ; ;