July 17, 1997: Morphed | July 16 | July 18 | 1997 | FOTD Home |

Fractal
visionaries:

Would you believe it! After all that fuss yesterday about the
innacurate terms in my XY-YZtest02 formula, I made a typo in the
would-be correct version. Since the innacuracy is so small
that
it will never be noticed, I'll let it stand as a monument to the folly
of laziness and haste. (For those who would like to see the
boo-boo, search for the ...88... that should be ...99... )

Today's formula might be described as an all-purpose one.
Highly
experimental, it does a little bit of everything, but nothing quite
right. Still, it draws some very unusual images.

By setting p1 to 0,0, 0,1, 1,0 or 1,1, all four odd planes can be
drawn. By setting p1 anywhere between these values, oblique
and
skewed planes can be drawn, though the angle of rotation is awkward and
difficult to determine.

The two parts of p2 draw planes parallel to the direction determined by
p1. P3 adds a variable portion of a second term to the
iterated
part of the formula. Real p3 determines the portion and imag
p3
determines the exponent.

Until today's fractal, I had been disappointed by the paucity of
midgets in the odd planes. I mean independent midgets not
buried
in the parent fractal. Of course, the Z^2 Mandelbrot set is
connected, and most likely the Z^2 julibrot is also connected, so
midgets standing all alone by themselves do not exist in the classic
set. But some Mandelbrot fractals are not connected, and
therefore have midgets standing alone, uncluttered by the clutter of
the parent fractal.

Today's image is part of the (Z^2+(0.2*Z^3)) julibrot. This
figure is one of these unconnected fractals. It is surrounded
by
a cloud of disconnected midgets like our galaxy is surrounded by
globular clusters. And these midgets are isolated and ready
to be
examined in all six planes. Today's image shows the central
part
of one of these outlying midgets sliced in the XZ direction.

The classic Z^2 julibrot is a parabola in the XZ plane, and all its
midgets are parabolic in shape. But in the XZ plane, the Z^3
julibrot is the "S" curve of the X^3 function, and in certain places
the features are virtually undistorted. When mixed with the
Z^2
figure, the Z^3 figure retains its "S" shape in the XZ plane, and when
offset, intersects the satellite midgets in very interesting slices.

Today's picture is part of a satellite midget of the (Z^2+(0.2*Z^3))
julibrot, sliced in the XZ plane. It resembles neither a
Julia
nor a Mandelbrot midget, but rather is like a morphed
combination. The finished image has been posted to a.b.p.f.
and
a.f.p. For tomorrow, I'll search out some even more
interesting
midgets in this fractal. Let's see if I can find something
unlike
anything seen before.

Jim Muth

jamth@mindspring.com

START 19.6 FILE=============================================

Morphed
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reset=1960 type=formula formulafile=basicer.frm

formulaname=SkewPlanes passes=1

center-mag=-3.54394/4.38209/58.54496/0.3982/90/3.8\

8578058618804789e-016 params=1/0/1.495/0/0.2/3

float=y maxiter=400 bailout=100 inside=0 logmap=yes

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frm:SkewPlanes {; Jim Muth

;p1=(0,0)=YW, (0,1)=XW, (1,0)=XZ, (1,1)=YZ

;p2=parallel planes, p3=optional extra term

a=real(p1), b=flip(cos(asin(real(p1)))), d=a+b,

f=imag(p1), g=flip(cos(asin(imag(p1)))), h=f+g,

z=real(pixel)+flip(real(p2)),

c=flip(imag(pixel))+imag(p2):

z=(d*(sqr(z)))+(real(p3))*(z^(imag(p3)))+(h*c),

|z| <= 36 }

END 19.6 FILE===============================================