Jim Muth wrote:
While toying with the Lyapunov Mandelbrot formula recently posted by Morgan Owens, I noticed that the formula produces images virtually identical to those drawn by the Fractint type=hypercomplex Mandelbrot formula. I have included two parameter files, one of which uses the Lyapunov Mandelbrot formula, while the other uses the hypercomplex Mandelbrot formula, yet the two files draw identical images.
Perhaps one of the math experts on the list can explain what is happening here. . . .
[snip...] (Not being a math expert by any means... ) Well, what Morgan's formula does is calculating two Mandelbrots where one of them is shifted relative to the other. Only those points which lie in the intersection of both sets will _not_ escape regardless if they are iterated in the z0- or in the z1- formula (so the iterated values for z0 and z1 will never move very far apart and the condition "bailout >= cabs(z0-z1)" will never be met). As for hypercomplex fractals, I have rendered a few in 3D over the years and these always looked like this: Imagine two cylinders (with the cross section of a "normal" fractal in the complex numbers) intersecting one another at right angles --> and this intersection volume actually being the hypercomplex fractal (or rather it's 3D-slice). Now we take a 2D-slice out of this object with the cutting plane oriented parallel to the complex plane: If the cutting plane _is_ the complex plane we get our usual complex fractal (duh!), that's right through the center of the intersecting cylinders, both cylinder's cross-section lie on top of each other. But if we move our cutting plane forward or backward from this position, the cylinders move away from each other and, since only the intersection of both cylinders really exists, our 2D-slice of the hypercomplex fractal gets smaller and smaller (just like the intersection of the two Mandelbrot sets in the formula below by setting bigger and bigger values for the "mag" parameter). (If anything above is not quite right or even plain wrong, I'd appreciate any corrections as long as those who speak up refrain from any flaming, bashing, kicking and whatnot... ;-)) Regards, Gerald ;---------------------- BEGIN PAR ----------------------------- For_Jim_Muth { ; reset=2002 type=formula formulafile=temp.frm formulaname=SuperposeMandel center-mag=-1/0/0.6666667 params=0/0/0/1/16/0 float=y outside=summ colors=@default.map } frm:SuperposeMandel {;params=arg/mag/bailout ; narg = real(p2) nmag = imag(p2) bailout = real(p3) z = c = pixel z1 = c1 = pixel + nmag*exp(flip(narg)) mxi = maxit - 2 esc = esc1 = exit = j = 0: IF (esc == 0) z = sqr(z) + c IF (bailout < |z|) esc = 1 ENDIF ENDIF IF (esc1 == 0) z1 = sqr(z1) + c1 IF (bailout < |z1|) esc1 = 1 ENDIF ENDIF IF (j >= mxi) exit = 1 IF (esc && esc1) z = -j ELSEIF (esc) z = 1 - j ELSEIF (esc1) z = 2 - j ELSE z = 3 - j ENDIF ENDIF j = j + 1 exit == 0 } ;------------------------ END PAR -----------------------------