The image in the banner at
the top of my webpage is taken from a famous fractal
known as the Mandelbrot set. The image is part of
the section of the Mandelbrot set for -1.187 < Re(z)
< -1.185 and 0.3 < Im(z) < 0.3015. The Mandelbrot
set is known as an iterative fractal as it is
generated from a recursive function that uses
complex numbers, *z*_{n+1} = *
z*_{n}^{2} + *c, *
where* z*_{0} = 0*. *The black
areas in the image correspond to values of *c*
where the iterated function stays bounded, while the
coloured areas correspond to values of *c*
where the function is unbounded (with different
colours corresponding to a faster rate of increase).
A fractal is a special type of
shape that is considered self-similar - it appears
the same at many different scales of observation.
The Mandelbrot set is not truly self-similar, but it
is possible to find many similar patterns at
different scales.
Fractals are of interest to
mathematical ecologists and biologists as
self-similar patterns often seem to occur in nature
in things like clouds, mountain landscapes, plant
structures and animal movement paths. |