The Fibonacci numbers and Mandelbrot’s fractals

The American-French mathematician Benoît Mandelbrot is very largely responsible for the interest in fractal geometry and the chaos theory that emerged in the 1980s.

Fractals are, firstly, only mathematically defined objects which are not one, two or three dimensional, but something in between. A fractal is therefore neither a whole line nor a complete surface, neither a complete surface nor a complete body. It is impossible to imagine that clearly. Nevertheless, objects occur in nature which come close to such fractals, for example a coastline, snowflakes or sponges. Structures can be found in them that repeat themselves and which are similar when considered approximately or in detail. That is why we also talk of self-similarity.

In the case of a snowflake, for example, all the six “points” have small six-pointed humps attached to all six points, on the points of which again correspondingly smaller six-pointed stars are developing – and so on. A sponge also shows a typical pattern of cavities and partitions in every stage of growth. An especially beautiful example of fractal geometry in nature is the Romanesco, a green variety of cauliflower.

Mandelbrot developed the concept of fractals to be able to describe these astounding characteristics of nature. But it was not so much the scientific significance as the possibility of producing pictures of great aesthetic appeal using simple algorithms and the computer that no doubt contributed to the popularity of fractals. The most famous of them is probably the Mandelbrot set. In it similar, but repeatedly new and incredibly beautiful structures are revealed in the area of their edges with every enlargement.

What have fractals to do with the Fibonacci numbers? The apple-shapes of the Mandelbrot set, which vary in size, arise in different periods of the repetition of mathematical algorithms. If these apple-shapes are examined it can be ascertained that between an apple of period 2 and an apple of period 3 the apple of period 5 is the largest. In exactly the same way the apple of period 8 is the largest between an apple of period 5 and an apple of period 3. And between the apple of period 8 and the apple of period 5 it is the apple of period 13.

The structures of the Fibonacci sequence and of the golden section seem to crop up everywhere in our surroundings. Although not all presumed connections have to be based on laws but may come about quite by chance, the significance of this mathematical principle is astonishing.