|
Fractal of the Day
by Jim Muth
|
|
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: 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 20.0 PAR-FORMULA 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 20.0 PAR-FORMULA FILE==================================
|
times.
