Fractal visionaries and enthusiasts:

At first glance, today's image is a disappointing and not very inspiring debris-filled Julia set, but it is an image that is intended to be tweaked, so don't hesitate to start changing the real(p1) and real(p2) parameters.

I have seen rumors circulating around the list lately that the fourth spatial dimension doesn't exist.   Of course it doesn't exist.   The orbits of the planets, among many other things, definitely obey the mathematically rigid inverse square rule of three-dimensional space.

But does this prove anything.   After all, we invented math ourselves and designed it specially to work with what we observe with our senses.

Does the fourth dimension exist then?   There is a big difference between not existing and being impossible.   There is nothing at all impossible about four-dimensional space with time as the fifth dimension.   There is no impenetrable barrier beyond the third dimension other than the inability of our minds to visualize spaces higher than the familiar everyday 3-D space.

One- two- and three-dimensional objects can be mathematically modeled and manipulated by computers.   With some additional complexity, four-dimensional objects can be modeled and manipulated by computers just as well.   The problem here is that a three-dimensional screen surface would be needed to properly display the resulting images of 4-D objects, and this 3-D surface would need to be viewed from the fourth dimension.   Since our minds have evolved to interpret our sensory input as a true image of a surrounding space having three spatial dimensions, we will never be able to visualize a fourth dimension, but this in no way prevents us from knowing what we would observe if we were able to do so.

Two of the more curious possibilities in 4-D space are absolute perpendicularity and double rotation.   Two planes are absolutely perpendicular when they intersect in a point, with every line in one plane perpendicular to every line in the other.   Double rotation exists when a 4-D object is subject to two independent rotations at the same time.   The points in the object move in a circular hyperhelix, somewhat like a slinky toy stretched out and curved into a circle with its ends connected.   When the two rotations are equal, as with today's scene, all the points except the stationary point at the center move in circular arcs.   Don't try to picture this motion.   I've been trying for years with no success yet.

In today's FOTD we have double rotated around the point at +003i in the north branch of Seahorse Valley, and stopped only 1/500th of one degree from the Julia direction, which is absolutely perpendicular to the Mandelbrot.   To see the 4-D double rotation inaction, decrease real(p1) and real(p2) toward 0 and 0, keeping the two parameters equal.   At (0,0) the Mandelbrot set will fill the screen, with Seahorse Valley at the center.   For a quick but dizzying trip through four-dimensional space, check the slices at (90,90) (89.998,89.998) (89.99,89.99) (89.96,89.96) (88,88) (82,82) (75,75) (62,62) (50,50) (30,30) (10,10).   Notice that the closer we come to the Mandelbrot orientation, the slower things go.   The final (10,10) slice is barely distinguishable from the actual M-set.

The scene may also be rotated from the Julia to the Mandelbrot by two simple 3-D type rotations.   Gradually reduce real(p1) to zero, which reveals the Oblate aspect of Seahorse Valley.   Then gradually reduce real(p2) to zero, which rotates the Oblate aspect back to the Mandelbrot.   Doing it this way passes through some quite interesting slices, while introducing quite a bit of stretching distortion.   Keeping the real(p1) and real(p2) parameters equal will prevent this distortion.   The passes=t option is needed to avoid major drop-outs.

Since the image is actually one in a series of images, I gave it no rating.   The calculation time of 2 minutes may be eliminated by visiting the FOTD web site at:where the finished image is posted.

Picture perfect weather prevailed here at Fractal Central on Tuesday, with digitally-enhanced blue skies, puffy clouds, and a temperature of 81F 27C.   Taking things for granted, the fractal cats slept through most of the day.   My day was once again average and uneventful.   The next FOTD trip to Seahorse Valley will be posted in 24 hours.   Until then, take care, and today's series of images could make an interesting animation, but it would require choosing some carefully measured increments.


Jim Muth
jamth@mindspring.com
jimmuth@aol.com

START PARAMETER FILE=======================================

Seahorse_Valley-24  { ; time=0:01:54.26-SF5 on P4-2000
  reset=2004 type=formula formulafile=basic.frm
  formulaname=SliceJulibrot4 passes=t float=y inside=0
  center-mag=0/0/0.7 maxiter=6400 periodicity=10
  params=89.998/0/89.998/0/-0.75/0.003/0/0/2/0
  colors=000QPGDC8MrzGdjBRV5DFTDsL9eE6S73E3gT2XL1ME0\
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  TxFfFFWVVMjjeaTSPJFPMEC9V2uK1aA0J3Xk2MW1BGmfuPLTKe\
  BFV8AL55A2DJw9Eh69U34FbaP }

frm:SliceJulibrot4    {; draws most slices of Julibrot
  pix=pixel, u=real(pix), v=imag(pix),
  a=pi*real(p1*0.0055555555555556),
  b=pi*imag(p1*0.0055555555555556),
  g=pi*real(p2*0.0055555555555556),
  d=pi*imag(p2*0.0055555555555556),
  ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
  sg=sin(g), cd=cos(d), sd=sin(d),
  p=u*cg*cd-v*(ca*sb*sg*cd+ca*cb*sd),
  q=u*cg*sd+v*(ca*cb*cd-ca*sb*sg*sd),
  r=u*sg+v*ca*sb*cg, s=v*sin(a),
  c=p+flip(q)+p3, z=r+flip(s)+p4:
  z=z^(p5)+c
  |z| <= 9 }

END PARAMETER FILE=========================================