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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: As the title of today's image implies, the image is impossible. With the value of real(p5) set to -2, the minibrots deep in the parent fractal must be of the order -2. Minibrots of a negative order are by definition impossible however. If they did exist, they would have a number of lobes 1 less that their order, but how can a minibrot have minus-3 lobes? This is what I decided to find out in the search for today's fractal, when I set the value of real(p5) at minus-2. The first problem was that, with such a large escape radius and a negative exponent of Z, the image turned out to be all blank inside stuff -- an evaporated image. This problem was quickly solved by rerendering the blank screen with an active inside fill. In this case I first tried the 'bof61' fill. When I did this, a ghostly arrowhead-shaped thing appeared, only vaguely resembling a Mandelbrot shape with a rudimentary East Valley. A closer search of this proto-East Valley revealed the symmetries that indicate minibrots are at the center. Zooming in on a prominent symmetry, I found today's scene, with its pseudo-minibrot at the center. The 'bof61' inside fill revealed the scene, but I found that the 'fmod' fill actually works best with this image. And what does a minibrot with minus-3 lobes look like? That question can be answered by checking today's image. I'm not sure that the thing at the center is a minibrot at all, but with its symmetrical surroundings, it sure does seem to be trying to be one. After all this mathematical excitement, I could rate the image no higher than a 6.5, 1/2 point of which is due to my coloring. The calculation time of a fireball 45 seconds will burn no one. And as always, the finished image is or soon will be available on the FOTD web site at: A perfect day here at Fractal Central on Saturday pleased humans and cats alike. The warm sun, blue sky and temperature of 77F 25C were perfect for everything but skiing. I doubt that anyone in the area had planned on skiing however. The day's FOTD might be a bit hasty, but the next FOTD is due in 24 hours. I suspect it will actually be ready on time, and it might even be a pretty good one. Until then, take care and make the impossible possible. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START PARAMETER FILE=======================================
It_Is_Impossible { ; time=0:00:45.13-SF5 on P4-2000
reset=2004 type=formula formulafile=allinone.frm
formulaname=DivideJulibrot passes=1 periodicity=10
center-mag=+0.21857780829516740/+0.051382415163788\
77/284576.4/1/-87.5/0 params=0/0/0/0/0/0/0/0/-2/0.5
float=y maxiter=1000 inside=fmod proximity=0.051
colors=000gaEhcEjeElgDniDokDjdLfZSaTZYMeTGlPAsrmmX\
Cn`DldDihEglFepFbtG`xHZzHXzKVyNUwPSccctVPsXOq_MpaL\
qbNrbPsbRtbTubVvcWwcYxc_UUPzcczddzdfzdhzdjzdlzdmze\
kzfjxfiughsggqhfoielicjjbhjafk`cl_alZ_mYYmX`kWciVf\
gUuuMppPlkRhfTdbV`YXXT_TPaPKcLFeF9eHBgIDhJFjKHkLJl\
MLnNNoOPpPRrQTsLXvOWuRVtUUtXTs_SsbRreQqhPqkOpnNpqM\
oPTGdPYsLooMlkMjgMgcNe_NcXN`TNZPOXLOUHOSBMOEOQHPSK\
QUNRWQSYTT_WU`ZVbaWddXfgYhjZjm_lvQHsWYnfvp`mqWerRY\
sMQtHImRHf`G_jFUsFVpIVnLVlNVjQVhSWfVWdXWb_W`aWZdMe\
NWXfOHBXL`iNzdOz_OyWOyRPxNPwIPwEQv9Qu5QuFRnPSgYTag\
UVcBgogHp`LpUPqOTqHXrA`r4ds6as7_s9YsAWtCTtDRtEPtGN\
tHLuJIuKGuMEuNC8zihVckTbmSaoR`rP_tOZvNYzN_xMYuLXrL\
VoKUlKSiJRfJPcIO`INDSkLWeT_``cWhgQpkLxoGzrBboUAmkE\
gWIbGKcHMcIOcJPcJRcKTcLUcLWcMYcN_cO`cObcPdcQecQWRQ\
MEQC2QD3PE4OF5OG6NH7NI8MJ8MK9LLALMBKNCKOEJOGJQIISK\
IUMHVOHXQHZSG`UGcWGcXFcXF }
frm:DivideJulibrot {; draws 4-D slices of DivideBrot Julibrots
pix=pixel, u=real(pix), v=imag(pix),
a=pi*real(p1*0.0055555555555556),
b=pi*imag(p1*0.0055555555555556),
g=pi*real(p2*0.0055555555555556),
d=pi*imag(p2*0.0055555555555556),
ca=cos(a), cb=cos(b), sb=sin(b), cg=cos(g),
sg=sin(g), cd=cos(d), sd=sin(d),
aa=-(real(p5)-2), bb=(imag(p5)+0.00000000000000000000001),
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)/(z^(aa)+bb)+c
|z| < 1000000 }
END PARAMETER FILE=========================================
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times.
