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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: The world of fractals holds many mysteries that will never be discovered, much less solved. But the mystery of today's image is one that has been discovered and, if I decide it is worth the effort, will be solved. At risk of being considered repetitive, today's image shows what happens to yesterday's image when the angle of the slice is double-rotated merely 0.0000685 of one degree from the Julia orientation. (Double rotation is a motion that can happen only in spaces of four or more dimensions.) When I began the rotation, I expected the octagon that appeared at the center of yesterday's image to become distorted. I did not expect it to break apart into two separate symmetrical holes shaped like near-perfect pentagons. But this is what happened. Now the question arises, which of the two pentagonal holes is a slice of the original rectangle that appears in the Mandelbrot aspect of the scene? And also, what happens to the other hole as we turn toward the Mandel direction? My intuition is that these questions are irrelevant, for there is only a single hole in the four-dimensional Julibrot figure, but this hole is shaped so irregularly that it can be intersected in more than one place by certain slices, such as today's slice. Switching the functions in today's formula shows a very distorted aspect of the hole that would seem to prove this. I named the image "Dual Pentagons". I was ready to rate it at another 8, but then I realized that it is basically the third identical image in a row, and decided that a rating of 5 is more appropriate. Those who enjoy the art side of fractals rather than the math side might be growing a bit bored. Those who are feeling the beginnings of boredom, as well as those who lack the time, may escape the 15-minute wait of rendering the image from the parameter file by downloading the completed GIF image from one of the FOTD web sites at: A really unusual thing happened Thursday here at Fractal Central -- for the first time in almost 4 weeks, it did not rain. True, a few showers were in the vicinity, but none reached F.C. The fractal cats, who have made up after yesterday's quarrel, were quite happy, and despite an oppressive temperature of 91F 33C, enjoyed several hours in the yard. Today is due to be a repeat, so I expect a repeat of the cats' good moods. Until next FOTD, take care, and it is sometimes easier to see the light when one is in the dark. Jim Muth jamth@mindspring.com jimmuth@aol.com |
START 20.0 PAR-FORMULA FILE================================
Dual_Pentagons { ; time=0:15:53.78--SF5 on a P200
reset=2002 type=formula formulafile=allinone.frm
formulaname=multirot-XY-ZW-new function=flip/ident
passes=1 center-mag=-0.00006842222800720/-0.000036\
15877643450/1535.55/1/-45/3.88578058618804789e-016
params=90.0000685/89.9999315/2.003/0/0/0/-1.743504\
25848689/6.977862219e-006 float=y maxiter=7200
inside=0 logmap=641 periodicity=10
colors=000voOtmMrkKpiKngJleJicJfaJd_JaYJZWJWTITRIR\
PIOMILKIIIIEFFGGIHHKJIMKIOMJQNKTOKVQLXRMZTN`UNcWOe\
XPgYPi_Qk`RnbSpcSreTtfUvgUxfXwfZve`vebueetdgtdisdk\
sblqalo`lm_llZljYlhWlfVleUlcTlaSm_RmZQmXOmVNmTMmSL\
mQKmOJmMImLKkOMiQOhSQfUSdWUcYWa_Y_a_ZdaXfcVheUjgSl\
iQnkPpmNrnLwoMtlMqiNneNk_NhVNeRMbNL_JLXFKUBJSCLRDO\
RERQFUQGXPH_PIdOJhOKnNLrNJwOLqNMlMOfMPaLQXLSTKTQKU\
NJWJJXGIYDIXGMWJQVMUUPYUSaTVeSYiR`mO`tRbpUdmXej_gg\
bhdejahkYkmVnnSqpPtqMwsJytGulNrdToXZlPdiHjfApdIocP\
nbWnacm`jm_qlZxl`wjawhbwfcvedvcevafv_guZhuXiuVjuTk\
tSltQmtOntNnkcnbslcrkcqjdpidohengemfelefkdfjcgibgh\
aggeaahXWkRQnMLrGFvB9z64s88iAC`CGSEKKFOIOKGWHEcEDk\
BClEClHBlKBlNAlQAlT9lW9lZ8la8ld7lg7lj6lm6lp6ls9kmC\
jhEibHhYJgTKeOLdJMbENa9O`4Rb7UdAWfDZhG`iJckMemPhoS\
kqVmrYpt`rvcuxfwyhuuetqbsn_rjYqgVpcSo_PmXNlTKkQHjM\
EiIChF9gB6f84eC8eFBmjSnjQ }
frm:multirot-XY-ZW-new {; draws 6 planes and rotations
;when fn1-2=i,f, then p1 0,0=M, 0,90=O, 90,0=E, 90,90=J
;when fn1-2=f,i, then p1 0,0=M, 0,90=R, 90,0=P, 90,90=J
a=real(p1)*.01745329251994, b=imag(p1)*.01745329251994,
z=sin(b)*fn1(real(pixel))+sin(a)*fn2(imag(pixel))+p3,
c=cos(b)*real(pixel)+cos(a)*flip(imag(pixel))+p4:
z=z^(p2)+c,
|z| <= 36 }
END 20.0 PAR-FORMULA FILE==================================
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