This file was moved to Wikimedia Commons from en.wikipedia using a bot script. All source information is still present. It requires review. Additionally, there may be errors in any or all of the information fields; information on this file should not be considered reliable and the file should not be used until it has been reviewed and any needed corrections have been made. Once the review has been completed, this template should be removed. For details about this file, see below. Check now!

Licensing

This work has been released into the public domain by its author, Gertbuschmann at English Wikipedia. This applies worldwide. In some countries this may not be legally possible; if so: Gertbuschmann grants anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.Public domainPublic domainfalsefalse

Summary

That our mapping ${\displaystyle f(z)}$ from the plane into itself is differentiable as a complex function, means that it is differentiable as a real function - that is, that its two components ${\displaystyle f_{x}(x,y)}$ and ${\displaystyle f_{y}(x,y)}$ are differentiable - and that these two components satisfy the Cauchy-Riemann differential equations:

${\displaystyle {\frac {\partial }{\partial {x}}}f_{x}={\frac {\partial }{\partial {y}}}f_{y}}$ and ${\displaystyle {\frac {\partial }{\partial {y}}}f_{x}=-{\frac {\partial }{\partial {x}}}f_{y}}$

if so, these two numbers are the real and imaginary part of ${\displaystyle f'(z)}$, respectively.

It is this condition that causes the characteristic features of the Mandelbrot and Julia sets for complex iteration. The usual family of iterations ${\displaystyle z^{2}+c}$ can (in coordinate form) be written ${\displaystyle (x,y)}$ → ${\displaystyle (x^{2}-y^{2},2xy)+(u,v)}$ (if c = u + iv), and if we here replace the y-coordinate of the function, that is 2xy, by 1.95×xy, the shapes in the Sea Horse Valley become distorted.

This thread-like and tattered look is typical for the real - or non-complex - fractals. For a function which is not, as in this case, the result of a mild interference in a complex function, the picture is often very chaotic, and the colouring can be impossible at most places, because our method of colouring presupposes that the sequences of iteration converge to a finite cycle, and for a non-complex iteration the terminus need not be a finite set. The terminal set is now called an attractor, and attractors can have very surprising shapes. Because of this, such an attractor is known as a strange attractor.

Original upload log

• 2010-04-21 09:44 Gertbuschmann 800×600× (431790 bytes) {{Information |Description = Non-complex Mandelbrot set |Source = I (~~~) created this work entirely by myself. |Date = ~~~~~ |Author = Gert Buschmann |other_versions = }}
• 2010-04-15 00:27 Gertbuschmann 800×600× (413599 bytes) {{Information |Description = Real fractal |Source = I (~~~) created this work entirely by myself. |Date = 16/4-10 |Author = Gert Buschmann |other_versions = }}