; July 18, 1997: Elephant Hints 
;
; elephant hints
;
; Fractal visionaries:
;
; Yesterday, I said that I would post something never before seen as
; today's fractal.  That's a rather hard promise to keep, but I think that
; I've come a bit close with today's picture.
;
; The two well-known planes have distinctive features, which become
; familiar with only a small amount of exploring.  The Mandelbrot planes
; have their familiar buds, valleys, stars, spirals, tendrils and of
; course, their midgets.  The Julia planes have everything the M-planes
; have except the midgets.  In addition, the Julia planes have those silly
; looking double spirals with the big noses, and are self-similar all the
; way down, which sometimes makes their exploration a bit dull.
;
; Until I found today's fractal, I considered the other four sets of
; planes rather dull also.  But I suspected that this couldn't be true,
; because everything of interest that appears in the Mandelbrot and Julia
; planes also appears in the other four sets of planes, but sliced in a
; different direction.
;
; Today's fractal is named for what it is -- the first interesting and
; unique object I have found in the XW plane.  As would be expected, it
; combines features of the Julia and Mandelbrot planes.  The chains of
; Mandelbrot buds twist around those silly big-nose, pop-eyed features so
; characteristic of Julia sets.  And everything is stretched and distorted
; as is always the case in the odd planes.
;
; Today's finished image has definitely been posted to a.b.p.f. and
; a.f.p.  I have already seen it there.  Anyone who can't find it on those
; groups has a problem with their news server.  I know that some servers
; are slow at picking up articles.  Sometimes it takes over a day for the
; Fractal of the day headers to appear on the AOL newsgroups list.
;
; One more thing needs to be discussed.  There is confusion as to which
; axes are the X, Y, Z, and W axes in these odd plane fractals.  I think
; of the Mandelbrot axes as X and Y, and the Julia axes as Z and W, while
; others consider the Julia axes as X and Y and the Mandelbrot axes as Z
; and W.  This situation leads to the confusion.
;
; But there is never confusion between the planes when the names
; Mandelbrot and Julia are used to describe them -- the names make the
; planes unmistakable.  By giving names such as Julia and Mandelbrot to
; the other four sets of planes, all this confusion could be avoided.  (I
; would suggest the Muth planes for one of the directions, but my humility
; prevents me from doing so. ;-)
;
; Anyway, while I'm considering names for the four odd sets of planes,
; I'll dump the cat out of the chair and ensconce myself in front of the
; TV to watch one of those great old sci-fi films.  By tomorrow, I'll have
; discovered other interesting scenes in the odd planes, and one of them
; I'll post.
;
; Jim Muth
; jamth@mindspring.com
;
; START 19.6 FILE=============================================

Elephant_Hints     { ; time=0:00:13.35-SF5 on P4-2000
  reset=1960 type=formula formulafile=basicer.frm
  formulaname=SkewPlanes passes=1 periodicity=10
  center-mag=+0.03329006362212719/+0.707306265659175\
  6/216.0686/0.15/92.296/71.545 params=0/1/2/-0.003/\
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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=proportional 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===============================================
;