Why do we need to calculate dimensions?

Question

After studying Kenneth Falconer’s Fractal Geometry, I got some techniques for calculating the Hausdorff dimension. However I am a little bit disappointed because in the application part, the book is still only discussing calculating dimensions. Always calculating dimensions — in number theory, in complex analysis, in thermodynamics, etc.

So here come two questions:

1. Why do we need to calculate dimensions? I mean, by calculating dimensions, can we get anything other than the values of dimension, such as a theorem in complex analysis (of which the statement doesn’t include the concept of Hausdorff dimension) that cannot be proved without using the properties of Hausdorff dimension?

2. Does the study of dimension give us a lot of unexpected result in physics? I know that the chaotic nature of a system makes its evolution unpredictable. (An example is forecasting weather.) But can fractal analysis make the weather forecast more accurate? Or does the study of fractals just illustrates the error we have when forecasting weather without helping us to minimize it?

I am having a hard time to make this question specific, but maybe you still think it's too general. I hope you can help me to make it more specific.

Answer 1

Why do we need to calculate dimensions?

The dimension is a useful characteristic of a fractal, just like the diameter or centre of mass are useful characteristics of normal geometrical objects or like the mean, median, variance, skewness, and so on are useful characteristic of a distribution:

• A fractal embedded in two-dimensional space with a dimension of 1.1 is considerably different than one with a dimension of 1.9. The former is almost a line, the second is more like a (slice of) cheese with holes.

• A coast line with dimension 1.1 is less ragged than one with dimension 1.9.

• The dimension of a vascular system can allow conclusions about its efficiency, robustness, and so on.

• A chaotic dynamical system with a dimension of 2.1 is clearly different from one with dimension 2.9. More specifically, if a continuous-time dynamical system with three dynamical variables has a dimension close to 2, you can expect that one of the degrees of freedom is almost spurious. For example, it could be that two of the dynamical variables are in some sort of synchrony most of the time. This is for example the case for the Lorenz oscillator with a dimension of 2.06.

Does the study of dimension give us a lot of unexpected result in physics? I know that the chaotic nature of a system makes its evolution unpredictable.( An example appears in weather report.) But does fractal analysis make whether report more accurate? Or does the sudy of fractals just illustrates the error we have in whether report, without helping us to minimize it?

First of all, dimensions are a way to detect chaos. For example, if you have a time series generated by some dynamical system, you can reconstruct its attractor (i.e., something having the same dimension) and measure its dimension – if it is significantly non-integer, you have a chaotic system.

However, if you have chaos, dimension (and other non-linear measures like Lyapunov exponents or entropy) does not really help you to better predict it, only to understand why there are problems predicting it.

The only use of fractal dimensions other as a mere measure that I am aware of is the theorem of Sauer, Yorke, and Casdagli, which makes an estimate of the sufficient embedding dimension for an attractor reconstruction based on the box-counting dimension and can work as a sanity check of your choice of the embedding dimension.

Answer 2

Consider the following problem in complex analysis: you have a compact set $K \subset \mathbb C$, and a bounded function that is holomorphic on $\Omega \backslash K$, where $\Omega$ is a domain containing $K$. Under which condition on $K$ can you automatically extend $f$ to a holomorphic function on $\Omega$? If it is the case for every such function $f$, $K$ is said to be removable.

Riemann's removable singularity says that you can do it if $K$ is finite. It is obvious that you cannot do it in general if $K$ has interior.

It turns out that the Hausdorff dimension of $K$ plays a role. If its Hausdorff dimension is less than one, $K$ is removable. If it is more than one it is not. It if is one... then it's complicated. See https://en.wikipedia.org/wiki/Analytic_capacity for more details and references.

As for the second question, I am not a physicist and cannot answer. I will simply say that I would expect that in questions related to numerical simulations, knowing the Hausdorff dimension would tell you what mesh size to use for a given precision.