The only "az^2+bz+c" variation I've seen is the "lambda map", f(z) = c z (1-z) = c z - c z^2. Here is an image: mrob.com/images/0-muency/lambda.jpg Notice the shape is different from the normal Mandelbrot shape, and they are related by "squaring" one to get the other. This version of the Mandelbrot set was actually published in Byte magazine[1] some months earlier than the Scientific American[2] that is normally thought to be the popular introduction of the Mandelbrot set. The Byte article was about fractals and simply showed a few pictures of the Lambda fractal, without a formula or other explanation. The "interior" was white, as in my image above, not black as the Mandelbrot set is more commonly drawn. I immediately recognized the importance of this new fractal and wished I could find out how it was calculated. I seem to remember reading somewhere that there aren't any other interesting shapes, but there must have been some qualification to that because you can get all sorts of shapes by applying a nonlinear transformation to the parameter, as in f(z)=z^2+1/c (see the "1/u plane" image at [3]) or f(z)=z^2+(K+exp(c)) (see my article on the exponential map, [4]) I guess the roots of f(z) are important, note that the lambda f(z) has roots at 0 and 1; the normal f(z)=z^2+c has a complex conjugate pair of roots +-sqrt(c) i. - Robert Munafo [1] Peter R. Sørensen. Fractals. Byte, Sep. 1984 p. 157. [2] A. K. Dewdney. Computer Recreations. Scientific American, Aug. 1985 Available online at: http://www.scientificamerican.com/media/inline/blog/File/Dewdney_Mandelbrot.... [3] http://www.matpack.de/Info/Mathematics/Fractals.html [4] http://mrob.com/pub/muency/exponentialmap.html On Thu, Jul 28, 2011 at 19:13, Dan Asimov <dasimov@earthlink.net> wrote:
Speaking of the Mandelbrot set, denoted by M:
The functions z^2 + c, c in C, contain just one examplar of each linear equivalence class of quadratic maps.
(I.e., in the set of maps
Q = {f(z) = a z^2 + b z + c with a,b,c in C and a != 0}
let f,g in Q be equivalent if there exists a linear map L(z) = d z + e with d,e in C and d != 0, such that Linv o f o L == g, where Linv is the inverse function of L.)
But {z^2 + c} is only one of many ways to parametrize this set of equivalence classes, and the choice of parametrization will drastically affect the appearance of the Mandelbrot set. [...]
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