Looking for a Specific Equation

[Ianpod2002]

Hi, I am sorry if I post this in the wrong section or it has already been answered.
I am looking for this equation I watched a program on a few days ago on PBS. I can't remember the name but will do my best to describe it.

It delt with a simple equation that lead to a picture with a hole on the center and many arms. If you zoomed in on any part of the picture it kept going on infinetly. each arm had branches that were described as 'hairs', and if you keep zooming in you eventually get to another hole that resembles the original except it is slighly twisted.

The equation was short, something like:
Z = x2 + c
the "=" is a two way door, so the equation goes into itself indefietly

Thank you very much for helping, I can't find it online and am very interested with it.
Ian

 

[Ianpod2002]

Oh, and I am looking for the name of the equation as well as the equation itself.
Thank you

 

[shmoe]

this sounds like the Mandelbrot set:

http://en.wikipedia.org/wiki/Mandelbrot_set
http://mathworld.wolfram.com/MandelbrotSet.html

and tons of other references as well as programs to generate these images can be found easily.

 

[HallsofIvy]

Oh, and I am looking for the name of the equation as well as the equation itself.
Thank you

I don't believe the equation has a specific name but I think the picture you are looking for is the "Mandelbrot Set" (although there is no "hole" in the mandelbrot set).

The equation, if that is correct, is z= z²+ c.

Here, z and c are complex numbers. Any complex number, x+ iy, can be thought of as representing a point (x,y) in the plane. Choose c= x+ iy. Starting with z₀= 0, form a sequence of complex numbers by "iterating" that equation- that is, do it over and over again:
z<sub>1[/sup]= z0²+ c, z2= z1²+ c, etc.


The point (x,y) is in the Mandelbrot set if and only if the sequence formed in that way, for c= (x,y), converges.

I might point out that Mandelbrot developed that in the 1960 working when computers had only very primitive graphics capabilities.

Even more remarkable, the Mandelbrot set is based on the "Julia" sets, developed by French mathematician Gaston Julia around 1900 when all calculation had to be done by hand!

The Julia set Jc, for a given complex number c, consists of all z0 so that the sequence formed above converges. Notice that the Mandelbrot set ranges through different choices of c while always taking z0= 0. The Julia set Jc has fixed c but ranges through different choices for z0.

Here's a nice website on the Mandelbrot set:
http://aleph0.clarku.edu/~djoyce/julia/explorer.html

Google on "Mandelbrot set" or "Julia set" for more.</sub>