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

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

Morphed            { ; time=0:00:11.97-SF5 on P4-2000
  reset=1960 type=formula formulafile=basicer.frm
  formulaname=SkewPlanes passes=1
  8578058618804789e-016 params=1/0/1.495/0/0.2/3
  float=y maxiter=400 bailout=100 inside=0 logmap=yes
  symmetry=none periodicity=10
  LITIWKShAbv1lt7krCjpIinNh }

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| <= 36 }

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