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Fractal of the Day
by Jim Muth
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Fractal visionaries and enthusiasts: Today's unusual midget takes us deep into the Zexpe fractal, a fractal that was named and explored by Lee Skinner. The word Zexpe is a contraction of Z exponent Epsilon, or Z^2.718... The formula Z^2.718+C draws a fractal that on the surface looks like a twisted version of the cubic Mandeloid, but inside is entirely different. Unlike the midgets of the Z^2+C Mandelbrot set, which always have the same basic shape, the midgets of the Zexpe fractal can apparently take most any shape whatever. Like all fractals with fractional exponents, the Zexpe fractal is not only infinite within its depths, its surface is infinite as well, taking the shape of an infinite hyperdimensional corkscrew. The MandelbrotBC formula, which drew today's image, calculates the more remote parts of this hypercorkscrew. The BC in the formula name stands for 'branch cuts', which are the discontinuities that, due to the multi-valued nature of the complex log function, must always appear in fractals with fractional exponents. The MandelbrotBC formula moves these branch cuts around as it slices through different parts of the corkscrew. It therefore can produce a great variety of different fractals from the same mathematical expression. The parent fractal of today's scene is a multi-lobed object with a small filament extending from the northernmost bud. The midget in today's picture exists in the spread-out East Valley of a prominent midget on this filament. I named the picture "A Zexpe Minibrot" for the obvious reason, and rated it a 7 because it actually is a rather striking scene. In fact it almost reminds me of a Hubble telescopic view of a planetary nebula. With a render time of 3-1/2 minutes, the parameter file is slow enough to make a download of the image file the better choice. The download may be found at: The fractal weather today here at fractal central was cloudy but milder. The temperature of 54F (12C) was just warm enough to satisfy the cats. It's now time to shut down the fractal shoppe for the night and give the cats their late night snack. Until tomorrow, take care, and brace yourself for more fractals. Jim Muth jamth@mindspring.com |
START 20.0 PAR-FORMULA FILE================================
A_Zexpe_Minibrot { ; time=0:03:31.02 -- SF5 on a P200
reset=2001 type=formula formulafile=branchct.frm
formulaname=MandelbrotBC passes=1
center-mag=+0.37032548119277480/+1.35149200434768100\
/526420/1/67.499 params=2.71828182845905/0/13.4/0
float=y maxiter=3000 inside=0 logmap=81 periodicity=10
colors=000A8F<3>ABIACJADKAEJ9HH8LF<3>4f93k72q51v4<3>\
ETFHMHKFKN8M<3>FNBDR8BV69Z37a1<3>DQSENZGKeHHlKFrUMtd\
SurZvwdwucvpbuibubat<4>QZrNYqKXq<3>AVo<9>Zkf`lecnd<3\
>lta<9>KqZHqZEqZ<3>3pY<3>abgj_jrXl<3>bR_ZQXVPURORPMU\
NKXKI_HHbDGeCFj<3>R9`V7YY6W<3>VBQUCPUDOTEMSFLSGK<8>P\
dGPgGPjF<3>OtE<3>cg_gdekajoZpsWuhYmZ_fUadOcbSe`WgZ_i\
4<2>kpUoraukZwdW<2>vLO<3>g`EddCYcBSeAMdAIc9Kb8Ma8<3>\
UiZWkdYme_oOaqe<2>gweiyz<3>qzirz5szOtz2<2>wzWxzC<2>z\
zZzz4<3>zzZzz2<3>zzTzz_zzz<3>zzlzzhzz4zzNzzOzzDzzNzz\
XzzA<3>zzTzzXzzazz7zzJzzVzzMzzTzz_zzWzz`zz2zzdzzP<3>\
zzbzzxzzqzzkzzpzzlzzS<2>zzQ
}
frm:MandelbrotBC = { ; Z = Z^E + C
e=p1
p=real(p2)+PI
q=2*PI*trunc(p/(2*PI))
r=real(p2)-q
Z=C=Pixel:
Z=log(Z)
IF(imag(Z)>r)
Z=Z+flip(2*PI)
ENDIF
Z=exp(e*(Z+flip(q)))+C
|Z| < 100
}
END 20.0 PAR-FORMULA FILE==================================
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