Sunday, 28 July 2013

Measuring the Complexity of Fractals

We don't need to convince anyone of the importance of fractals to science. The question we wish to address is measuring the complexity of fractals. When it comes to more traditional shapes, geometries or structures such as buildings, plants, works of art or even music, it is fairly easy to rank them according to what we perceive as intricacy or complexity. But when it comes to fractals the situation is a bit different. Fractals contain elaborate structures of immense depth and dimensionality that is not easy to grasp by simple visual inspection or intuition.

We have used OntoNet to measure the complexity of the two fractals illustrated above. Which one is the most complex of the two? And by how much? The answer is the following:

Fractal on the left - complexity = 968.8
Fractal on the right - complexity = 172.9

This means that the first fractal is about 5.6 times more complex. At first sight this may not be obvious as the image on the right appears to be more intricate with much more local detail. However, the image on the left presents more global structure hence it is more complex. The other image is more scattered with smaller local details and globally speaking it is less complex . This means that it  transmits less structured information, which is precisely what complexity quantifies. Finally, below we illustrate the complexity map of the fractal on the left hand side.