Dr Edward A. Codling

 

Lecturer in Mathematical Biology

Departments of Mathematical Sciences & Biological Sciences

University of Essex, UK

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Fractals

 

 

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, zn+1 = zn2 + c, where z0 = 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.

The above images were generated using a simple program written to run in the R language. This has not been optimised and may take a while to run, particularly if you change the resolution to be high.

 

R Code (source file) for Mandelbrot set

 
   

Page last updated August 2010