Today's post is a 3D image of a Chebyshev fractal. The Chebyshev method is similar to the Halley method and is used to iteratively solve for roots of equations. The fractal is colored based on how quickly the root is found. The method is from Computers, Pattern, Chaos, and Beauty by Clifford Pickover, page 277. It was challenging to make a good model of this fractal since it has very thin walls and deep cuts. It took Fracton 99 hours and 31 minutes to make the model on my 2 core PowerPC Mac Pro. The 3D image is rendered in Cheetah 3D.
Here is a link to a web page with an image:
I will also list the PAR file here for people collecting them from the mail list:
Chebyshev { ; Exported from Fracton.
reset=2004 type=formula formulafile=fracton.frm
formulaname=F_20120816_1336 passes=1 float=y
center-mag=-5.306250000000001/-22.9666015625/0.853\
3333439999999/1/14/0
params=7/0/0/0/0/0/0/0/0/0 maxiter=100 inside=-1
proximity=0
colors=UzJ<13>B`3<4>H1C<3>i6X<2>znP<5>zS7<222>znP }
frm:F_20120816_1336 {
; Type: Chebyshev
; From Computers, Pattern, Chaos, and Beauty by Clifford Pickover
; See p277 for a related image and formula derivation
z=pixel:
lz=z,
; Function f(z) = 64*z^7-112*z^5+56*z^3-7*z
f=64*z^7-112*z^5+56*z^3-7*z,
f1=64*7*z^6-112*5*z^4+56*3*z^2-7,; First derivative
f2=64*42*z^5-112*20*z^3+56*6*z,; Second derivative
z=z-2*f*f1/(2*f1*f1-f2*f),
abs(|z|-|lz|)>=0.001
}
--
Mike Frazier
www.fracton.org
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