FOTD -- July 03, 2009 (No Rating)
Fractal visionaries and enthusiasts:
There are two versions of today's scene, calculated by two
different formulas, one sliced in the Mandelbrot direction, the
other sliced in the almost-Julia direction, give or take a
fraction of one second of arc. The fast Mandelbrot version
takes under 3 minutes to calculate, the slow Julia version takes
4-1/2 hours. Since the images are virtually identical, I advise
running the fast version and taking my word that the slow
version appears identical. The slow version is the topic of
this discussion however.
In three-dimensional space two planes must intersect in a line,
and lie at a certain single unique angle to each other. In four-
dimensional space two planes intersect in only a single point
and lie at two independent angles to each other. When the two
angles are equal, every point in one plane in a circle around
the point of intersection is at the same distance from the other
plane. It makes no sense to a 3-D programmed mind, which
visualizes a conic section, but that's the way it is.
Today's fast image intersects the Julia plane at a double angle
of 90,90 degrees. The slow image intersects the Julia plane at
a double angle of a tiny fraction (about 1/50th) of one second
of arc from the actual Julia direction, yet the change from
Mandelbrot to Julia features has not yet appeared. The change
will appear in the next 0.000001 degree of double rotation
however. Some pretty abstract thinking is needed to figure out
what's happening here. (It's beyond me!)
All this abstraction prevents me from rating the image, though
the calculation time of the fast version of under 3 minutes
makes it relatively simple to see the image. Masochists who
want to assure themselves that both images are virtually
identical may start the slow version and wait half a day or so.
The best way to see and compare both versions is to view them on
the FOTD web site at:
<
http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
Partly cloudy skies and a temperature of 82F 28C made Thursday
near ideal here at Fractal Central. The fractal cats enjoyed
things as much as they could, while my day was average.
The next FOTD is due to be be posted in 24 hours, but we expect
a big rush job on Friday, so don't be surprised if the FOTD is
late. Until whenever, take care, and keep searching.
Jim Muth
jamth@???
jimmuth@???
START PARAMETER FILE=======================================
Fast_Version { ; time=0:02:50.81-SF5 on P4-2000
reset=2004 type=mandel center-mag=+0.2500000999954\
752/-0.00000000004996773/8.672315e+011/1/-165/0
params=0/0 float=y maxiter=500000 inside=0
symmetry=xaxis periodicity=0 mathtolerance=0.05/1
colors=000EpADq9Cs8d4ge5fe6ef8ef9dfAcgBbgCahEahF`h\
G_iHZiIYjKYjLXkMWkNVkOUlQUlRTmSSmTRjMLlQOmURoXUp`X\
rcZsgaujdvngzujxsiwqiuoitmhrkhqjhohhnfgldgkbgiagh_\
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AE8AF8AH9BI9BKABLACNACOBCQBDRCDTCDUCEWDEXDEZEF_EFa\
EFbFGdFGeGGgGGhGGjEFlDEnC }
Slow_Version { ; time=4:31:16.93-SF5 on P4-2000
reset=2004 type=formula formulafile=basic.frm
formulaname=SliceJulibrot4 center-mag=-0.002030815\
15044265/-0.0022062339955831/15067.5/1/-105/0
params=89.999999/0/89.999999/60/0.2500001/0/0/0/2/\
0 float=y maxiter=500000 inside=0 symmetry=xaxis
periodicity=0 mathtolerance=0.05/1
colors=000d4ge5fe6ef8ef9dfAcgBbgCahEahF`hG_iHZiIYj\
KYjLXkMWkNVkOUlQUlRTmSSmTRjMLlQOmURoXUp`XrcZsgaujd\
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BI9BKABLACNACOBCQBDRCDTCDUCEWDEXDEZEF_EFaEFbFGdFGe\
GGgGGhGGjEFlDEnCEpADq9Cs8 }
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=========================================
