"M-set anatomy" remake
There are several reasons to remake old Java based pages
JS is easy-to-use because to make rather complex scripts you need not compiler but
just browser and text editor.
- 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.
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 more restricted than Java and therefore safer
(e.g. JS can not write files to your hard drive).
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.
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).
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"
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.
Two more fractal images
Next: Rotation Numbers and Internal angles of the M-bulbs
updated 8 Sep 2013