Why fractals?

A significant part of TA, if not all, is based on Averages, and as such, it relies heavily on the Gaussian (or Normal) Distribution which is the statistical translation of the Random Walk Theory.
Indeed for Averages (and that includes all kind of Moving Averages, Simple or Exponential) to really be as meaningful as TA considers them, prices variation must actually be described adequately by the Gaussian Distribution and its counterpart in random process, the Brownian Motion.

It is interesting to note that there is a contradiction inherent to the practise of TA. In his "Technical Analysis of the Financial Market", John Murphy wrote (with good reason):
The Random Walk Theory (...) claims that price changes are "serially independent" and that price history is not a reliable indicator of future price direction. In a nutshell, price movement is random and unpredictable(...) It also holds that the best market strategy to follow would be a simple "buy and hold" strategy as opposed to any attempt to "beat the market."

Something I completely agree with, but then, if a technical analyst is to reject this Random Walk view of price movement, shouldn't he reject as well the mathematical ramifications of this assumption rather than to use them as tools.
In a Gaussian model, the average (the mean) clearly is a good information to consider, it is the quantity that has the highest probability to be realised, and the closest to the average, the higher the probability is.

The large pool of experimental data we have from financial markets, however, tells us that they don't follow a Gaussian distribution, they diverge from it in various ways but a remarkable one is that they are fat-tailed , which means that the probability for the variable to be far away from the average is actually higher than in the Gaussian model (i.e. extreme variations are more frequent than what is predicted by the model). And that is important, because it tends to make our beloved Average less useful, in terms of prediction, while the differences are not such that Averages don't retain any usefulness. But more precise tools may likely be derived from a more fitting model of the real price movement.

Another problem with the Gaussian model is that it assumes continuity and evenness of change. Benoit Mandelbrot in "Fractals and Scaling in Finance" wrote:
In the classical (Gaussian) theory of errors, a large change would typically result from the rare chance simultaneity of many large contributing causes, each of them individually negligible. In economics, this inference is indefensible. Typically, the occurrence of a large effect means that one contributing cause, or at most a few turn out ex-post to be large.

This non-evenness, as well as the discontinuity of price movement (which is obvious given the structure of the process of price determination, the apparent continuity is just an artefact of price representation), contribute even further to undermine the validity of information given by Averages and even more so, by Moving Averages.

Mandelbrot again, remarks:
In particular, price continuity is an essential (but seldom mentioned) ingredient for all trading schemes that prescribed at what point one should buy on a rising market, and sell on a sinking price. Being discontinuous, actual market prices will often jump over any prescribed level, therefore, such schemes cannot be implemented.

Then, what are the alternatives to the Gaussian Distribution ?
Mandelbrot goes on discussing a few of them in his above-mentioned book. I won't do that here. The alternative I wish to discuss on this blog is the one most promising, in terms of modeling the behaviour of price movement, as far as I know.
It is the option involving the use of fractals. The models developed with fractals have so far shown a better fit than Gaussian models (as well as other alternatives), and I therefore hope that they can lead to the development of more efficient TA tools than the ones existing today.