tip1-300
Bengt continues his deepzooming explorations. This time at the very
tip of the mandelbrot in the (-2.0,0.0) region. Here is a small
set of preview size zooms leading to a 640x480 size image. I'll let his
text lead you through his exploration from here.
/ng
The following seven deep-zoom images are centered around a very small mini-MS in the usual Mandelbrot set, located at a distance of approximately 8.6 * 10^(-306) from the tip of the main antenna, at -2.
To find the mini-MS I made a zoom of 10^300 times around the tip at -2 by means of a .par-file, containing { center-mag=-2/0/1e300 }:
tip1-300
Any (high) magnification around -2 gives essentially the same picture, but at the nodes between the colored bubbles small mini-MS:s can be expected. I centered a zoom-box around the leftmost node visible. After roughly one hundred zooms the following picture emerged at magnification 2.96*10^451, showing the first signs of a mini-MS.
tip1-451
Then, for the next hundred zooms or so, an increasing number of radial, colored fields appeared:
tip1-453
tip1-535
tip1-584
Magnifications in these pictures were 1.72*10^453, 2.28*10^535, and 6.94*10^584, respectively. I have chosen these three as typical examples of new features emerging, essentially the bifurcation of radial fields, and the appearance of circular fields mixed with the radial ones. The colors inside the sectors are, however, changed by color cycling. This mixed circular/radial structure is more clearly seen in this picture:
tip1-600
(magnification 1.24*10^600) where one might also vaguely discern a mini-MS. Magnifying another hundred times I finally arrived at this, maybe the smallest mini-MS, ever seen:
tip1-602 click on the image to get the 640x480
version.
The magnification is 2.09*10^602 , maximum number of iterations 60000, and the full size calculation took 800 hours on my 90 MHz Pentium computer. Try downloading it, load it into Fractint, look at the parameters and cycle the colors!
The (prolonged) circular structures are predominant, and the radial patterns much less distinct than in the earlier published 10^240 times magnifications. Also note the tendency to horizontal patterns just to the right of the center of the image. The picture has a lot of intricate details although it is not artistically very spectacular. I would say that it gives more mathematical ideas than most pictures at lower magnification, maybe because it looks in some sense simpler. One might wonder how much of the details are artifacts from the finite precision of computer calculations.
A final note on the location of the mini-MS:
The image-file shows a truncated value x-value for the center of the image, but saving the parameters into a parameter file, one obtains the corners with full precision. The par-file for the last image is also available.
The following image is a zoom in 164 times on an antenna
in the upper part of the TIP1-602 image. It shows a
mini-MS, off the symmetry line, the smallest non-symmetric
mini-MS I've seen.
tip1-604
The picture is only in preview because of the immense calculation time. On my Pentium 90 MHz the time to calculate this small picture was nearly 300 hours.
"bengtmn@algonet.se"
You can also view more detailed information on Fractint's Deepzooming by looking at the online documentation.