At 03:51 PM 10/31/11 -0600, Mike wrote:
....Anyone that has taken electrical engineering is familiar with the formula for parallel resistance:
rp = 1 / ( 1 / r1 + 1 / r2 )
If you put a mandelbrot equation at r1 and r2 with a different center point you have it....
Mike: Very good idea -- parallel resistance. Electricity and fractals. I guess it's all one grand unified universe, based on numbers. What your formula creates appears to be fractals of the type created by combining positive and negative exponents of Z, such as the one created by the parameter file at the end of this letter. Increase the value of real(p2) and the Mandies shrink; decrease the value and they expand and merge, as they are starting to do in the image drawn by the included file. I also see fractals drawn by your formula that resemble those created by my DivideBrot series of formulas, with debris filling the usually empty insides. But your formula gives control over the position of the Mandies that I cannot achieve with my formulas. I'll have more to say later. Jim Muth jamth@mindspring.com START PARAMETER FILE======================================= Multi_Mandies { ; time=0:00:23.29 SF5 at 200MHZ reset=2004 type=formula formulafile=basicer.frm formulaname=MandAutoCritInZ function=ident inside=0 center-mag=-0.414649/-0.442222/0.9690955/1/-35/0 params=1/2/-0.025/-2/0/0/0/0 float=y maxiter=1500 colors=00000e0e00eee00e0eeL0eeeLLLLLzLzLLzzzLLzLzz\ zLzzz000555888BBBEEEHHHKKKOOOSSSWWW___ccchhhmmmsss\ zzz00zG0zV0zj0zz0zz0jz0Vz0Gz00zG0zV0zj0zz0jz0Vz0Gz\ 00z00zG0zV0zj0zz0jz0Vz0GzVVzbVzjVzrVzzVzzVrzVjzVbz\ VVzbVzjVzrVzzVrzVjzVbzVVzVVzbVzjVzrVzzVrzVjzVbzhhz\ lhzqhzuhzzhzzhuzhqzhlzhhzlhzqhzuhzzhuzhqzhlzhhzhhz\ lhzqhzuhzzhuzhqzhlz00S70SE0SL0SS0SS0LS0ES07S00S70S\ E0SL0SS0LS0ES07S00S00S70SE0SL0SS0LS0ES07SEESHESLES\ OESSESSEOSELSEHSEESHESLESOESSEOSELSEHSEESEESHESLES\ OESSEOSELSEHSKKSMKSOKSQKSSKSSKQSKOSKMSKKSMKSOKSQKS\ SKQSKOSKMSKKSKKSMKSOKSQKSSKQSKOSKMS00G40G80GC0GG0G\ G0CG08G04G00G40G80GC0GG0CG08G04G00G00G40G80GC0GG0C\ G08G04G88GA8GC8GE8GG8GG8EG8CG8AG88GA8GC8GE8GG8EG8C\ G8AG88G88GA8GC8GE8GG8EG8CG8AGBBGCBGDBGFBGGBGGBFGBD\ GBCGBBGCBGDBGFBGGBFGBDGBCGBBGBBGCBGDBGFBGGBFGBDGBC\ G000000000000000000000000 } frm:MandAutoCritInZ {; Jim Muth a=real(p1), b=imag(p1), d=real(p2), f=imag(p2), g=1/f, h=1/d, j=1/(f-b), z=(((-a*b*g*h)^j)+(p4)), k=real(p3)+1, l=imag(p3)+100, c=fn1(pixel): z=k*((a*(z^b))+(d*(z^f)))+c, |z| < l } END PARAMETER FILE=========================================
What your formula creates appears to be fractals of the type created by combining positive and negative exponents of Z, such as the one created by the parameter file at the end of this letter. Increase the value of real(p2) and the Mandies shrink; decrease the value and they expand and merge, as they are starting to do in the image drawn by the included file. I also see fractals drawn by your formula that resemble those created by my DivideBrot series of formulas, with debris filling the usually empty insides.
Thanks for the comments Jim. The image with your parameter file does look very similar to the one I posted. I have spent many many hours exploring with your MandAutoCritInZ formula. To be honest, that was the main reason I am adding arbitrary precision math to formulas in Fracton. I wanted to see what lies deep into some of those images. I have found some things but the best images so far are still at depths that work with regular double precision math. The arbitrary precision math is hundreds of times slower than double precision because of the z ^ b power function. I found that if I limit my search to integer powers and substitute z * z * z ... into the formulas it is only about 20 times slower. Thats doable if you are patient and don't go too deep. I did have one question I wanted to ask about your formula. I am guessing the auto crit part is the z = (((-a*b*g*h)^j) + p4. It isn't obvious to me what that does. Is there a simple explanation? -- Mike Frazier www.fracton.org
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Jim Muth -
Mike Frazier