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.

The parameter file for the fractal is:

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
 }

Export Model Settings:

Model Type: Height Field
Grid Samples X 1000
Grid Samples Y 1000
Min Ext Angle 10.000000
Min Edge Len 0.100000
Min Face Area 80.000000
Model Size X 10.000000
Model Size Y 5.000000
Model Size Z 10.000000
Upper Clamp 30.000000
Lower Clamp 14.000000
Texture Vert: Color Index

Model Statistics:

Total vertices: 715974
Total faces: 1073685
Total polygons: 138
File size: 55407909
Time 99 hours 31 min on a 2 core PowerPC Mac Pro