## "M-set anatomy" remake

There are several reasons to remake old Java based pages
• The old version was optimized for 1024×800 screen resolution. New pages are fitted to your browser window (reload page after resizing window). It is not clear yet if this fitting suits well for HD displays but simple (non fractal) boundary length depends on image size only linear.
• JavaScript just in time compilers are very efficient now (as like as Java ones). Therefore it is not necessarily to support in mobile browsers two very similar technologies (iOS and Android do not support Java applets).
• Multi-touch controls are added for mobile devices. Use two fingers to drag or zoom an image (scroll pages by one finger). One single touch is equivalent to mouse click.
JS is easy-to-use because to make rather complex scripts you need not compiler but just browser and text editor.
JS is more restricted than Java and therefore safer (e.g. JS can not write files to your hard drive).
See also An Interactive Introduction to Splines remake.

## CPU vs. GPU

Modern GPUs generate fractal pictures very quickly. See e.g. WebGL based Julia sets animation, animation with the Distance Estimator algorithm or Makin's fractal animation. But we can use double floats only in experimental WebCL yet.

Unfortunately direct calculation of all pixels is not power efficient. The boundary tracing algorithm is 4-8 times more efficient for the standard Mandelbrot and Julia sets. It is rather complicated and suits more for CPU. 4 JS workers are used for direct calculations.

# The Mandelbrot and Julia sets. Introduction.

We start with a Quick Tour around the Mandelbrot and Julia sets. If you are not bored to death you can proceed with the rest.

The Mandelbrot set is made by iteration of the complex map (function) zn+1 = zn2 + C for different C and zo = 0. Points on the parameter plane C with bounded zn form the Mandelbrot set (the black region on these pictures). Color outside the Mandelbrot set shows, how fast zn go to infinity.
The Mandelbrot set contains small copies of the "main cardioid" (in the white square) shown on the right picture. These small copies are connected with the main cardioid by filaments which are formed by other tiny cardioids. These strucrures are called the "Mandelbrot hair" or filaments. Use two fingers to drag or zoom an image (scroll the page by one finger). One single touch is equivalent to mouse click. Click mouse to zoom in (controls by mouse).

Each point C in the parameter space specifies the geometric structure of the corresponding Julia set J(C) in the dynamical plane z. If C1 is in the Mandelbrot set, the Julia set is connected. If C2 is not in the Mandelbrot set, the Julia set is a Cantor dust. You see connected 'Douady's rabbit' and 'Cantor dust' below.
 You see that every J set is self-similar. But the most amazing thing is that the M's filaments and corresponding J set are similar too (see the right top and bottom images). And two more similar M and J sets below.

Contents     Next: Rotation Numbers and Internal angles of the M-bulbs
updated 8 Sep 2013