FOTD -- June 24, 2009 (No Rating)
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-dimentional space with time as the
fifth dimension. There is no impenetrable barrier beyond the
third dimension other than the inability of our minds to visual-
ize 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 manipula-
ted 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 pre-
vents 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 per-
pendicular to the Mandelbrot. To see the 4-D double rotation in
action, 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) para-
meters 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:
<
http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
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@???
jimmuth@???
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 center-mag=0/0\
/0.7 params=89.998/0/89.998/0/-0.75/0.003/0/0/2/0
float=y maxiter=6400 inside=0 periodicity=10
colors=000QPGDC8MrzGdjBRV5DFTDsL9eE6S73E3gT2XL1ME0\
B7I_F9I7AG3581NPbRRlhJCM96TYzBWv8Og5GT28ExsjheZUSN\
FEBfHIWCDL89A44u0yRvByT8UU2FF13Nj1BNGwvpR4QD2HFteh\
3SU2EF1wfLhWFULAFA5JsHC`B6I5W9_O6RG4I8294K02A09uV6\
aK3JAsYZ`MNIBBBvO7bG3J8a_UOyoJd9OHiC8NvmObXGJG8LfU\
np5PQ2XjKMVDBF6TACF5f72LW3_keid`CKU4AF2pMLQBAQy_Ji\
RDVI6F9UXwKMcABK49C268134rp5db3RQ2DD1J64C42621iu`f\
OiWIYLCNA6BjVkZN_NFOB7C5pIR2GI1A905RDxI8c94Kx81U40\
gEZDPT8GJ489zP9eG6L83p8`b6RQ4ID29hRrUI_F9IdYRtEqa9\
_J4IqUY_KMIABfaEWSALJ7A93e1WV0OL0GA08qSMREBxWnUGPZ\
k1HO0wEYc9MK4B7507pV1gR0MDEFDuMbTBJUGFKAAA55tlkikT\
NOEkYqOHRtZaeQSSHJE89VsnNeaFSP7ECWtoGSQqqY__MIIBHS\
S8EE1HK08As9pS4QVI7KC4A62sQK`HDI86wSGcIAK95gCOT8GE\
48Xes`zzXfrTNjQ4c9XZZWLxV7qcOjldctukYnrChkP`e`UZlM\
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=========================================
