Something to Ponder During Your Next Acid Trip

http://www.youtube.com/watch?v=RsC-TK8MPc8

The Mandelbrot Set is the set of values of c in the complex plane for which the orbit of 0 under iteration of the complex quadratic polynomial z_{n+1}=z_n^2+c remains bounded. That is, a complex number c is part of the Mandelbrot Set if, when starting with z0 = 0 and applying the iteration repeatedly, the absolute value of zn remains bounded however large n gets.

The Mandelbrot Set M is defined by a family of complex quadratic polynomials P_c:\mathbb C\to\mathbb C given by P_c: z\mapsto z^2 + c, where c is a complex parameter. For each c, one considers the behavior of the sequence (0, P_c(0), P_c(P_c(0)), P_c(P_c(P_c(0))), \ldots) obtained by iterating P_c(z) starting at critical point z = 0, which either escapes to infinity or stays within a disk of some finite radius. The Mandelbrot Set is defined as the set of all points c such that the above sequence does not escape to infinity.

Correspondence between the Mandelbrot Set and the bifurcation diagram of the logistic map The intersection of M with the real axis is precisely the interval [-2, 0.25]. The parameters along this interval can be put in one-to-one correspondence with those of the real logistic family, z\mapsto \lambda z(1-z),\quad \lambda\in[1,4].\, The correspondence is given by c = \frac\lambda2\left(1-\frac\lambda2\right). In fact, this gives a correspondence between the entire parameter space of the logistic family and that of the Mandelbrot Set. The area of the Mandelbrot Set is estimated to be 1.50659177 ± 0.00000008.

Mandelbrot Set, you're a Rorschach Test on fire

You're a day-glo pterodactyl

You're a heart-shaped box of springs and wire

You're one badass fucking fractal

And you're just in time to save the day

Sweeping all our fears away

You can change the world in a tiny way

Jonathan Coulton

"Mandelbrot Set"