I like fractals. Complexity arising from simple mathematics and all that. This is my blog and I can ramble on about whatever I want so lets talk about fractals today, or rather one fractal in particular.
Pick a complex number z0. Repeatedly run this number through the transform zi+1=zi2+z0. You'll find that for some choices of z0 this number runs off to infinity, for others it will spiral and shoot around, but never get very large. If you colour in spots on an Argand diagram for which this number never gets large then you get the following picture:
That's right! It's your friend and mine Mr. Mandelbrot. The set that contains all of the complex numbers for which zi+1 = zi2+z0 never goes to infinity.
So far so good, it's nothing you haven't seen before.
Now, I found a while ago about a more interesting way to plot this. The traditional way of viewing the Mandelbrot set immediately discards any points that run off to infinity, but what do they do before they get there? do they shoot straight off, or do they spiral around, gradually moving out.
Well lets have a look, here I plotted out the paths of every point that ends up going to infinity. The brightness of each pixel represents the number of times it was hit.
click for full image
Isn't it pretty! This way of plotting the Mandelbrot set is called the Buddhabrot. (IDL or MPI FORTRAN code available upon request)
Pick a complex number z0. Repeatedly run this number through the transform zi+1=zi2+z0. You'll find that for some choices of z0 this number runs off to infinity, for others it will spiral and shoot around, but never get very large. If you colour in spots on an Argand diagram for which this number never gets large then you get the following picture:
That's right! It's your friend and mine Mr. Mandelbrot. The set that contains all of the complex numbers for which zi+1 = zi2+z0 never goes to infinity.
So far so good, it's nothing you haven't seen before.
Now, I found a while ago about a more interesting way to plot this. The traditional way of viewing the Mandelbrot set immediately discards any points that run off to infinity, but what do they do before they get there? do they shoot straight off, or do they spiral around, gradually moving out.
Well lets have a look, here I plotted out the paths of every point that ends up going to infinity. The brightness of each pixel represents the number of times it was hit.
click for full image
Isn't it pretty! This way of plotting the Mandelbrot set is called the Buddhabrot. (IDL or MPI FORTRAN code available upon request)
