Almost everyone learns growing up that a line is 1 dimensional, a plane is 2 dimensional, and the space we live in is 3 dimensional. One day, a crazy theoretical physicist came along and argued that the universe consists of 4-dimensional space/time. That physicist’s name was Albert Einstein. A slightly less well-known name is Benoit Mandelbroit. Mandelbroit was a mathematician who argued something much more profound: dimensions are not discrete. The standard example of this phenomenon involves measuring the coastline of Britain. A Euclidean scientist would think that measuring Britain’s coastline with a smaller, more refined, ruler would give a more accurate measurement than a larger, less refined ruler would, and that with a small enough ruler, the length of Britain’s coastline could be determined to any degree of precision. This is actually not the case!

In fact, if the coastline of Great Britain is measured in units of 100 km, then its coast is roughly 2800 miles long. However, if our Euclidean scientist tries to refine this measurement by using units of 50 km, then the coast is 3,400 km long. That’s an increase of 600 km!

The reason for this apparent paradox is that unlike a boring line which has no new structure as you zoom in, Britain’s coastline has detailed structure on all levels of magnification. This new structure makes Britain’s coastline a naturally occurring fractal.

A true fractal is an object which is self-similar, meaning that as you zoom in on the fractal, you see smaller copies of it.

Examples of fractals abound. Three particularly interesting ones are: Pre-modern cities, the cells that make up the human body, and the stock market.

In pre-modern cities, particular cities which grew “naturally” out of collections of settlements, each group of people living in the city would want to have everything they need close to themselves. This

The fact that the cells that make up the human body are fractals is being used in cancer research. Any biology student would understand that the surfaces of cancer cells display different proteins than the surfaces of regular cells. Recent research has shown that cancer cells have a different fractal dimension than regular cells. This can be used to help better identify cancer cells and better design drugs targeted to the specific structure of the cancer cells.

The stock market is a very interesting example of a fractal because unlike the other two examples where fractals were used to find structure in pre-modern cities and normal cells, the fractal nature of the stock market means uncertainty. Fractals are not smooth, they jump around a lot on small distances. According to Nassim Taleb, this can be used for a different investment strategy. Instead of trying to predict which stocks will do well based on their history on the market, Taleb proposes investing small amounts in many different high-risk stocks. You will loose some money here and there, but all you need is one fractal spike on the market and you could make millions!

Believe it or not, your resting heartbeat is also a fractal and over time will naturally get to around 120 bpm without your noticing.