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The Fractal Geometry of Nature, Benoit Mandelbrot

My interest in this book is that the author and founder of the theory of fractal geometry described it fairly recently in a paper as the essential introductory text in the area. Fractal geometry is basically an attempt to define mathematically the difference between complex data at separate scales. So if you imagine the coastline of a country, then measure that coastline with a 1 meter ruler and you will get a shorter distance than if you measure it with a 1cm ruler, and again with a 1mm ruler etc etc. It’s taken me a while to read because, having not studied maths for a while the terminology is quite alien, the concepts are simple enough but what I’m really looking for is the tools used to define the differences between scale and so I want to understand the maths properly.

The central importance of fractals to the study of cooperation (or indeed any other social construct) is that it allows you a potential way round a central issue in economics which is that social data is not linear in nature. My interest in looking into psychology and complexity is in seeing how other subjects tackle this problem. To illustrate the point, if you recall the research methods lecture with the economics professor we had the other week he was attempting to model migration patterns by reducing migration data to single values and then add in probabilistic margins for error. This is a typical way for an economist to tackle a complex area, you might have seen Mervin King from the bank of England giving his economic forecast today along the same lines – we know the data is complicated, we have introduced these assumptions and we believe this is a sensible percentage error. Is their another more accurate way to model this complex phenomena?

The approach Mandelbrot takes is to find a scale and measure it as accurately as you can with the tools available, then take as many more scales as you can and calculate the trend of increased or decreased complexity between the scales. This is fundamentally different from what you do in economics, in economics you scale your data up in a linear fashion (e.g. you calculate GDP by adding up the domestic output for each industry sector) which doesn’t account for that scale may be a factor in itself, you then add in a margin for error to account for the fact that you cant understand how the interactions between the actors who produce your data may change with scale.

Now, it’s fair to say that if your data is linear – i.e. one change always has one effect on one thing then there will be no difference in approaches but if by changing one thing has an effect on several things which has an effect on several more then the linear approach quickly gets to rather large margins for error rather quickly. The fractal approach doesn’t.

To summarise, fractal geometry is about understanding the changing shape of data at scale which captures the relationships between many data points. So if you know accurately what happens at one scale and the fractal dimension your data scales by you can predict accurately what your data will look like at larger scales or visa versa.

I’m now going to go look how a psychologist/sociologist attempts to quantify the phenomena they observe as a comparison. Any psychologist want to point out a book for me?

Written by Paul on November 10th, 2010

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