# Fractal

From a precious metals investor’s point of view, fractals are complicated structures of prices of metals (for instance, charts of prices of metals). The complexity of these structures (charts) can be described by a mathematical measure – the fractal dimension.

### Eric asks:

Eric, Jill and John meet at Jill’s house to go to a football game. They still have some time so Jill offers them a cup of tea and they talk and drink in the living room.

#### Eric, the Beginner

#### Prof. Jill, the Investor

#### Eric, the Beginner

#### Prof. Jill, the Investor

#### Eric, the Beginner

#### Prof. Jill, the Investor

#### Eric, the Beginner

#### Prof. Jill, the Investor

#### Eric, the Beginner

#### Prof. Jill, the Investor

#### Eric, the Beginner

#### Prof. Jill, the Investor

#### John, the Trader

#### Eric, the Beginner

#### John, the Trader

The higher values the fractal dimension reaches, the more complicated the structure (the chart) is. If one remembers that the charts tend to be relatively simple during a rally or a decline (the prices tend to follow a trend line and a line is a relatively simple structure) and relatively complicated in proximity to a top or bottom or during a sideway trend (we all know the jagged chart lines near bottoms or tops – they tend to be complicated), it turns out that the fractal dimension **might indicate a possible top, a possible bottom or a sideway trend** (this is the case, when the values of the fractal dimension are relatively high).

On the other hand, relatively low values of the fractal dimension indicate a possible rally or a possible decline. So in this sense, fractals might help you to identify possible turning points in the price paths of metals.

## What is a fractal?

We have briefly explained how fractals might be of help to precious metals investors. Now, let us explain what is usually described by the term *fractal*.

A fractal is a complicated structure that consists of parts which are similar to other parts of this structure. This is best to explain on an example. Please look at the picture below.

This is a moderately complicated structure consisting of triangles of different sizes. After a few seconds, one might get the impression that some parts of the biggest triangle are similar to the others. Let us zoom in on the structure and chose one of its parts (let us call it part A – picture below).

Now, let us take a look at a slightly bigger part of the whole structure (let us call it part B – once again, on the picture below).

It is easy to notice that the two mentioned parts are similar to each other – for instance each of them has a white triangle in its middle. What is more, if we compare parts A and B to the whole structure, it turns out that both of the parts are similar to it as it also has a white triangle in the middle of it. These observations lead us to the conclusion that the whole structure is a fractal. As a matter of fact, it is and it is called the Sierpinski triangle.

A situation, in which a part of an object is similar to other parts of it or (and) to this object as a whole is called self-similarity. So, we would be able to say that the triangle on the pictures above is self-similar. If an object is self-similar than we may infer that it is a fractal.

It might seem that fractals are a purely theoretic concept – **how similar triangles would be able to help one make an investment decision?** However, in reality fractals have a lot to do with the precious metals market. To explain that let us take a look at the annual chart for gold from 2010.

At first there is no clear indication that any parts of the chart are similar to each other. However, if we zoom in on the chart, namely on September, October and November, the situation turns out to be different.

On both of the charts the first half of the time span displays a period of growth, which is followed by a relatively sharp decline right in the middle of the time span, which decline is in turn followed by a period of growth in the second part of the time span. This second period of growth has a quite bumpy ending on both of the charts. These observations lead us to the conclusion that the charts are similar to each other. If we remember that the lower chart is in fact a part of the upper chart, we might say that charts of gold have parts which are similar to other parts of these charts – they are self-similar. Such a remark implies that **charts of gold might be fractals**.

If charts of gold are fractals, then it is possible to measure their complexity using measures typical for fractals. One of these measures is the aforementioned fractal dimension. As we have explained before, the fractal dimension can be used to identify possible tops or bottoms from past prices of metals (*hmm… the chart line for gold has been quite jagged recently and the fractal dimension is pretty high… am I facing a top right now?...*). So, if charts of gold are fractals and the fractal dimension might be used to measure the volatility of the market, it seems that fractals might help **anticipate possible tops and bottoms**, which is of extreme importance for any gold investor.

The points raised above suggest that the analysis of fractal behavior of the prices of gold is important for precious metals investors. However, it might seem complicated to determine whether current prices are similar to prices from the past from different time spans and therefore it is relatively hard to determine whether the prices have a fractal structure. What is more, the calculation of the fractal dimension also proves to be fairly complicated. Because of this, fractals are not frequently used among precious metals investors even if they might be of help.

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