
Mandellambda Sets

formulas: mandellambda


This type is the "Mandelbrot equivalent" of the lambda set. A comment is
in order here. Almost all the xmfract "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
xmfract are the results of mistakes! Nevertheless, thanks to Kevin for
keeping us accurate!

(See description of "initorbit=" command in Image Calculation Parameters
for a way to experiment with different orbit intializations).

