Why should anyone care about computing the Hausdorff Dimension?

Question

I am trying to justify an thesis/investigation that will (hopefully) lead to a formula for computing a particular class of Cantor-like sets. The question that I have not satisfactorily answered, according to my department chair, is 'Why should anyone care about computing the Hausdorff Dimension?'

Answer 1

For one, it gives an invariant which helps us distinguish sets which are otherwise rather difficult to tell apart, and in a way which is reflected in actual, interesting properties of the sets.

On the other hand, there are many situations in which the Hasdorff dimension of a set controls analytical properties of solutions of certain differential equations and that sort of thing —imagine you are trying to solve a PDE on an open set with a very complicated boundary, for example (and no, this is not done out of pure pleasure: «real life» problems lead to to such monstruosities). This is quite useful when it is needed!

Answer 2

I may be shunned by real mathematicians for this but I think it can provide some detail about "how long a coastline is" since they exhibit fractal behaviour. The length keeps getting longer when you measure more closely.

Numberphile did a good video on it https://www.youtube.com/watch?v=7dcDuVyzb8Y

Answer 3

For instance, there are various results about Kleinian groups and the hausdorf dimension of their limit sets

http://en.wikipedia.org/wiki/Kleinian_group