;     Date: Mon, 16 Feb 2004 10:54:10 -0500
;     From: Jim Muth <jamth@mindspring.com>
;  Subject: [Fractint] FOTD 16-02-04 (Triple Point [7])
;       Id: <1.5.4.16.20040216105646.299fc3d0@pop.mindspring.com>
; ---------
; 
; FOTD -- February 16, 2004 (Rating 7)
; 
; Fractal visionaries and enthusiasts:
; 
; This happens every time I start thinking about the fourth 
; dimension.
; 
; Today's image starts what may prove to be a lengthy journey in 
; the oblique slices of the four-dimensional Z^2+C Julibrot figure.
; 
; The formula I use for this exploration was developed about five 
; years ago by John Goering and revised slightly by me.  I have 
; not posted this particular version previously.  Since I will be 
; using it often in the days to come, be sure to remove the frm: 
; from its name and copy it into your formula file.
; 
; This formula does not draw every possible slice through the four-
; dimensional figure, but it draws more slices than any other in 
; my vast collection.  It can do a smooth rotation from the Julia 
; to the Mandelbrot aspect.  And it reveals one of the least known 
; oddities of the world of fractals -- that there are two slices 
; through the Julibrot that produce a Mandelbrot set.  One of 
; these slices is drawn when the (p1) and (p2) parameters of the 
; formula are set to zero.  I leave it to the intrepid fractalists 
; to find the parameters that reveal the second Mandelbrot set.  
; One hint -- the (p1) and (p2) parameters need to be varied only 
; somewhere between +1 and -1.
; 
; Today's image is another view of the orange midget of 
; yesterday's image, using the same color palette.  I have named 
; it "Triple Point".  There is not much to be said about it, 
; except that it is a remote slice of the Julibrot, which appears 
; as though it is being pulled in three directions.  The rating of 
; 7 and render time of 7 minutes give an overall value of 84.
; 
; Starting with tomorrow's FOTD, I'll do some heavy-duty 
; exploration of the fourth dimension.  But before that time 
; arrives, save a few minutes by downloading the finished image 
; from Paul's web site at:
; 
;        <http://home.att.net/~Paul.N.Lee/FotD/FotD.html>
; 
; Sunday was sunny here at Fractal Central, but the temperature 
; hovered around freezing all day.  This was too chilly for the 
; fractal felines, who spent the day indoors, seeking the warmest 
; spots they could find.  Thomas managed a few hits at his old 
; playtoy before he settled for the day.  Tippy just settled.
; 
; Today is starting even colder, with a temperature of 14F -10C.  
; The cats, who are growing slower every year, will likely have 
; another inactive day.  As for me, when the work is finished, 
; I'll have a fractal day.  Until next time, take care, and don't 
; get dizzy trying to figure those Julibrot rotations.
; 
; 
; Jim Muth
; jamth@mindspring.com
; jimmuth@aol.com
; 
; 
; START PARAMETER FILE================================

Triple_Point { ; time=0:07:11.60--SF5 on a P200
  reset=2003 type=formula formulafile=julibrot.frm
  formulaname=SliceJulibrot passes=1
  center-mag=+0.00000000000130513/+0.000000000008816\
  47/2.524132e+011/0.03598/101.821445919962812/-76.3\
  113167840462694 params=0.83333/0.5/0/0/0.351424051\
  7840664/0.06386667204878391/0.3514240517840664/0.0\
  6386667204878391 float=y maxiter=5000 inside=0
  logmap=-820 periodicity=10
  colors=00001P02P03P14Q25Q36Q47R58R69R7AQ8BQ9CPADPB\
  EPCFPDGQEHRFIQGJRHKSILTJMUKNVLOWMPXNQYNRXOSXPTXQUX\
  RVXSWXTYXT_X`bageenkjuonvhgwa`wWUxPNyIGyCAxFBxHCxK\
  DxMDwPEwRFwTFwWGwYHv`IvbIveJvgKviKueJtbIs_IrXHqUHp\
  QGoNGnKFmHFlEEkBEfDLbERZGXVHbQIeMKgILiEMgCOeBPd9Rc\
  8SbAUcCVcEWcGYcIZcK_dMadObdQcdSfdUheWjeYme_oe`meXk\
  iTilPgpLesHcwDZztiTqgVofXleYjd_hc`ebbcaca`eZ_gXZhU\
  YjSXkQWmNVnLUpJTqLVmMWjNYgPZdQ``RaYTbVUdSVeOXgLYhI\
  ZiFXjHWjIVjJUjKTjLSjMQkOPkPOkQNkRMkSLkTPgSTdRWcQ_c\
  PbcOfcNicMmcLpcKtcJwcJscHocFkcEgcCdcA`c9Xc7Tc5Qc4O\
  cBMcIKeOIgVGi`EkgCpmAtt9xzCvuEtpGrkIpgKobNqYPsTRuP\
  TwKVxFXzBZzL`zVbzc`z_ZzWXzSVzPTzLRzHQzEQzIQzLPzPPz\
  SPzWPzZPzbPzeRzcSzbUz`Vz_XzZYzX_zW`zVazTczSdzQfzPc\
  zO`zVgzUdzYaz`ZzcXzeVzdTzcVzbYza_zaaz`cz_fzZhzYjzY\
  lzXozWqzVszUuzUnzSgzQ`zOUzMOzLVzJ`zIgzHmzGtzFzzEwz\
  LtzRrzXozblzhjznczVezThzR }

frm:SliceJulibrot    {; draws most oblique slices
  pix=pixel, u=real(pix), v=imag(pix), a=pi*real(p1),
  b=pi*imag(p1), g=pi*real(p2), d=pi*imag(p2), 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=sqr(z)+c
  |z|<=9 }

; END PARAMETER FILE==================================
;