Way down a Seahorse Valley!

from Bengt Månsson

The following picture is from the "Seahorse Valley" of the largest mini-MS (at about -1.75), just 10^(-6) above the real axis. It is a deep zoom in the sense that the maximum number of iterations is 3.2 million, and these are needed. Decreasing to 1.6 million makes details vanish. However the magnification is "only" 1.54*10^10, which means that the floating point processor is in use. Yet the calculation time was almost a week on a 90 MHz Pentium.

MIN2-10.GIF click on the image to get the 800x600 version.

The spirals curl up to something resembling the outgrows on the main components. At lower resolution these curls become black and one might think that the valley is narrower than it really is. To a smaller degree this "curling up" phenomenon is also found at lower magnification and also in the valley at -0.75.

The following image is a zoom in 87 times on the large sprout in the lower left part of the above image. Notice the striking resemblance to a Julia set!

MIN2-12.GIF

Another 12 times magnification resulted in this:

MIN2-13.GIF

It is interesting to see how the curling up in the first image reappears here. There is a mathematical theorem stating that a certain part of a Julia set in general resembles a part of the Mandelbrot set, at the corresponding c-value. But what we see in the last two images is something else. Whole Julia-like structures, with details similar to typical outgrows on components of the Mandelbrot set. Obviously, something is going on "behind the scene". I wonder what.

Mathematical note:

or, approximately,

-1.768529152467685161553845318849198520183563232421875

(on the real axis). From this, and an estimation of the sizes of the main components enclosing the valley, a suitable starting point and magnification can be calculated. Then it just requires minor justifications to get a valley intersection filling the screen.

*************************************

mini1_2              {
  reset=1920 type=mandel passes=1
  corners=-1.7685291525900553/-1.7685291524167748/9.999351580606626e-007/1\
  .0000651184459504e-006 params=0/0 float=y maxiter=3200000 inside=0
  colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzzzLzzz000555<3>HHHKKKOO\
  OSSSWWW___ccchhhmmmssszzz00z<3>z0z<3>z00<3>zz0<3>0z0<3>0zz<2>0GzVVz<3>zV\
  z<3>zVV<3>zzV<3>VzV<3>Vzz<2>Vbzhhz<3>zhz<3>zhh<3>zzh<3>hzh<3>hzz<2>hlz00\
  S<3>S0S<3>S00<3>SS0<3>0S0<3>0SS<2>07SEES<3>SES<3>SEE<3>SSE<3>ESE<3>ESS<2\
  >EHSKKS<2>QKSSKSSKQSKOSKMSKK<2>SQKSSKQSKOSKMSKKSK<2>KSQKSSKQSKOSKMS00G<3\
  >G0G<3>G00<3>GG0<3>0G0<3>0GG<2>04G88G<2>E8GG8GG8EG8CG8AG88<2>GE8GG8EG8CG\
  8AG88G8<2>8GE8GG8EG8CG8AGBBG<2>FBGGBGGBFGBDGBCGBB<2>GFBGGBFGBDGBCGBBGB<2\
  >BGFBGGBFGBDGBCG000<6>000
  }

The following image is a zoom in 8 times on the second lowest colored "bubble" to the left in the first image (MIN2-10.GIF) above.

MIN2-11B.GIF

The maximum number of iterations is as high as 10^8 in order to avoid large unresolved spots.

After another zoom a mini-MS appeares, the first one I have seen in this deep region of the valley. The magnification is 157 from the image MIN2-10.GIF.

MIN2-12B.GIF

The mini-MS is located roughly in the middle of a "bridge", somewhat to the right and up from the center of the "bubble". The "bridge" connects complicated areas and is bounded at its sides by more uniformly colored areas. The scenario somewhat resembles that of the double spirals higher up in the valleys, where mini-MS:s are found. Indeed, I found the mini-MS by looking for such similarities. "bengtmn@algonet.se"

You can also view more detailed information on Fractint's Deepzooming by looking at the online documentation.

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