 Height field 3D model of Chebyshev rendered in Cheetah 3D

 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