2.9 Mandellambda Sets
(type=mandellambda)
This type is the "Mandelbrot equivalent" of the lambda (p. 49) set. A
comment is in order here. Almost all the Fractint "Mandelbrot" sets are
created from orbits generated using formulas like z(n+1) = f(z(n),C),
with z(0) and C initialized to the complex value corresponding to the
current pixel. Our reasoning was that "Mandelbrots" are maps of the
corresponding "Julias". Using this scheme each pixel of a "Mandelbrot"
is colored the same as the Julia set corresponding to that pixel.
However, Kevin Allen informs us that the MANDELLAMBDA set appears in the
literature with z(0) initialized to a critical point (a point where the
derivative of the formula is zero), which in this case happens to be the
point (.5,0). Since Kevin knows more about Dr. Mandelbrot than we do,
and Dr. Mandelbrot knows more about fractals than we do, we defer!
Starting with version 14 Fractint calculates MANDELAMBDA Dr.
Mandelbrot's way instead of our way. But ALL THE OTHER "Mandelbrot" sets
in Fractint are still calculated OUR way! (Fortunately for us, for the
classic Mandelbrot Set these two methods are the same!)
Well now, folks, apart from questions of faithfulness to fractals named
in the literature (which we DO take seriously!), if a formula makes a
beautiful fractal, it is not wrong. In fact some of the best fractals in
Fractint are the results of mistakes! Nevertheless, thanks to Kevin for
keeping us accurate!
(See description of "initorbit=" command in Image Calculation Parameters
(p. 126) for a way to experiment with different orbit intializations).