Mandellambda Sets

Nice Image: Mandel-lambda
Mandel-lambda type

This type is the "Mandelbrot equivalent" of the lambda 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 the description of the "initorbit=" command in Image Calculation Parameters for a way to experiment with different orbit intializations).
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