Saturday, June 5, 2010

Julia sets and more fractals

Following on from my last post on the Mandelbrot set I've created a Julia set generator. Julia sets are a similar sort of fractal to the Mandelbrot set. The Julia sets I am showing are all generated using the function zn+1 = zn2+c. Thus for every possible value of c we get a different Julia set. In this viewer the Julia sets are shown in red. There are many other types, but these are the classic Julia sets that led to the development of the Mandelbrot set.

The Julia sets come in two varieties; connected and disconnected sets. Either all the points in the Julia set are connected to each other (forming a single central entity) or all the points are disconnected (forming a Cauchy set, also called Fatou dust). The Mandelbrot set is generated from this fact, basically the Mandelbrot set is all the values of c such that the generated Julia set. In fact the Mandelbrot set forms a pictographic index of all the connected Julia sets. If you zoom in enough into the Mandelbrot set you will eventually find a copy of every conceivable Julia set. Which is pretty cool.

Some of the Julia sets form incredibly beautiful shapes, while others can be pretty mundane, I have put a list of some interesting sets below the viewer, just click on the links and the values will load into the viewer. Otherwise please explore the space and try your own values. If you find a combination of values that looks particularly impressive please leave the values as a comment for other people to try. The instruction for use are the same as the previous Mandelbrot viewer with the addition of being able to change the values for c by giving the real and the imaginary parts.


Some interesting sets

Click links to load:

Thursday, June 3, 2010

The Mandlebrot Set

So I have been reading allot about fractals lately (you should start with this cool book), and of course the Everest of fractals is the Mandelbrot set. The Mandelbrot set is (for those of your who don't know) an intriguing mathematical object, but I won't go into the math of it. Just take my word for it that this set is one of the coolest things ever discovered) Visually though it has boundary that is infinitely complicated, containing myriads of twisting patterns and curves (in fact in a sense it contains every possible curve somewhere on it boundary. You can zoom in on the curve and it will remain detailed at any level of zoom, never becoming a simple shape.

I've also been wanting to try out the HTML5 canvas for a while, so I decided to make a Mandelbrot set viewer with javascript. doing something like this in a browser would have been impossible a few years ago, but now with modern browsers (this excludes Internet Explorer, sorry this wont work for you but I don't care) even computationally difficult tasks are becoming feasible using javascript.

Anyway,play around with it, maybe you get a little interested, maybe you check out some of the ideas. Maybe not.


  • Below is the Viewer (If you don't see a cool shape it probably doesn't work on your browser, you should upgrade).
  • You are going to want to zoom in on a specific spot on the boundary of the black object in the middle (which is the Mandelbrot set), to do this click and drag around the place you want to zoom in on.
  • Eventually when you zoom in too far the shape will get boring, this is simply because the computer only renders an approximation. You can increase the precision by increasing the precision value in the text box and pressing the reset button. The higher the precision the longer in takes to render the picture.
  • Pressing the reset button will also let you zoom all the way out and start exploring and new part of the set.