What is a "fractal"?

Fractal is a word coined in 1975 by Benoit Mandelbrot. It is formed from the Latin "fractus", which means crushed, broken, broken.

In the language of mathematics, a fractal is a set with the property of self-similarity. In other words, each element of the set is an exact or approximate copy of the whole.

It seems that this is a purely mathematical concept, but in reality we are faced with fractals every day. Trees, mountains, plants, and even the circulatory system have a fractal structure.

Fractals in nature

As you know from the course of geography, each of the countries has its own area of ​​territory and length of the border. If the country is washed by any sea or ocean, then it also has a sea border. But how is the length of these boundaries determined?

In 1967, Benoit Mandelbrot posed the following question: "What is the length of the UK coastline?"

It turned out that it is impossible to correctly answer this seemingly simple question.

The length of the coastline can only be measured approximately. As you zoom out, more and more small headlands and bays need to be measured. And since there is no limit to scale down, then the length of the coastline has no limit. That is, it is infinite.

The English mathematician Lewis Fry Richardson faced the same paradox in 1951, he noticed that Portugal defined its land border with Spain at 987 km, while Spain, in turn, at 1214 km.

The solution to this problem is possible by setting the minimum distance. For example, if the length of the coastline is measured in kilometers, then small bends less than one kilometer in length are not counted.