Mandelbrot Set — ever heard of it?
Math can be very beautiful…and the Mandelbrot set proves this!
This is what the Mandelbrot set looks like.
It’s even more fascinating when you get to know the mathematics that goes behind it.
The Mandelbrot set is basically a set of complex numbers.
The given figure represents a complex plane with a complex number z=x+iy plotted in it.
The distance of z from the origin(0+i0) is called as the “magnitude of z”, represented as |x+iy|.
Now, what is the connection between the Mandelbrot set and the complex plane?
Let’s consider a complex number c and a function fc(z) associated with it. fc(z) is defined as fc(z)=z²+c, i.e, it takes some complex numbers z as it’s input and outputs some complex numbers z²+c. We analyze the behavior of 0 while iterating fc.
i.e, when z=0 and c=1:
f1(0) = 0 + 1
f1(1) = 1+1 = 2
f1(2) = 4+ 2 = 6 and so on….
The Mandelbrot set is concerned with distance or the magnitude of these numbers in the complex plane to the origin(0+i0).
There can be 2 cases here,
i) the size of the complex numbers (i.e, the distance of the complex number from the origin goes on increasing so much that it blows up!) or
ii) the size of the complex sequence remains bounded(less than 2 to be precise).
The Mandelbrot set thus defines the set of complex numbers that don’t blow up. Thus every point within the Mandelbrot set should be within the boundary 2. The points on the edges of the set represent uncertainties, i.e, these points bounce between blowing up and not blowing up.
That was a very basic intro about Mandelbrot set. If you have read until this point, please feel free to drop your comments or suggestions or useful links on Mandelbrot set.
Thank you! Have a great day.
