DEFINITION OF FRACTAL

Fractals are geometric entities whose shape does not vary as the length scale varies.

Basically, if a small portion of the fractal object is enlarged, using an appropriate scale factor, structural characteristics appear, which exactly reproduce those of the object itself when it is not enlarged. The characteristic whereby successive enlargements of small portions of the object always show the same structure is called self-similarity.

The term fractal derives from the Latin "fractus" (broken), as well as the term "fraction"; therefore, from a mathematical point of view, fractals are objects of fractional dimension (not integer), between 0 and 3. It is possible to have fractals of dimension 1.5 or 2.4 or even 0.3.

The concept of fractional dimension was introduced in 1918 by the German mathematician of Jewish origin Felix Hausdorff (1868 - 1942), for this reason the "fractal dimensions" are often called "Hausdorff dimensions".

In 1975, the Polish mathematician, physicist and engineer naturalized French Benoît Mandelbrot (1924 – 2010) introduced, for the first time, the term "fractal". The Mandelbrot set is one of the most famous fractals:

Mathematical representation of the Mandelbrot set.

The "fractal shape" is extremely jagged, contrary to what happens in Euclidean geometry for the elementary figures, which have regular shapes and lose their structure when they are observed using small scales.

The non-Euclidean geometry, which studies the fractal structures, is called “fractal geometry”.

A typical example of the application of fractal geometry is represented by the study of coastal strips, since the determination of the effective length of this strips is a very complex task. In fact, the measurement of the length of a particularly jagged coast, as the precision of the measurement increases, provides structures similar to those encountered using smaller and smaller scales. Furthermore, the length of the coastal strips grows without ever converging to a well determined value.

VON KOCH CURVE

The Helge von Koch curve or "von Koch's snowflake" is one of the first fractal curves of which a description is known. It appeared for the first time in 1904, in a document written by the Swedish mathematician Helge von Koch (1870 - 1924) entitled "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire".

The first three iterations of the von Koch curve are shown in the figure below:

Fig.1

The recursive procedure that allows to geometrically construct the Helge von Koch curve is summarized in the following steps:

1. start from an equilateral triangle with side equal to 1;

2. each side of the equilateral triangle is divided into three segments of equal length;

3. the central segment of each side of the equilateral triangle is replaced by two other segments, which have the same length as the eliminated segment and they are arranged so as to form an angle of 60⁰, as highlighted in the image at the center of Fig.1. Therefore, on each side of the starting equilateral triangle there will be 4 segments. The figure thus generated corresponds to a six-pointed star, formed by a total of 12 segments;

4. the procedure just described is applied to each of the 12 segments that form the six-pointed star, as highlighted in the right-hand image of Fig.1;

5. by repeating the same procedure n times, with n tending to infinity, successive figures will be obtained with increasingly jagged outlines;

6. the final figure is the fractal.

All the steps, which precede the construction of the complete fractal, are called "pre-fractals".

The peculiarity of the von Koch curve is its infinite perimeter, while the area enclosed in it is finite.

The fractal dimension of the von Koch curve is given by the relation:

D = lnN/lnR = log4/log3 ~ 1,262

where appears the general expression of the "dimension according to Hausdorff", that is:

D = lnN/lnR

N is the number of pieces into which the fractal can be broken, so that each piece is similar to the initial fractal.

R is the reduction factor (in the case of von Koch's snowflake the sides are divided by 3 each time).

The dimension of von Koch's snowflake is intermediate between a line and a surface. Closer the dimension is to 2, more the fractal will tend to uniformly cover a surface. Instead, as the dimension approaches 1 the fractal will look more and more like a simple line.

FRACTALS AND NATURE

The nature that surrounds us provides some objects with shapes approximately similar to those of fractals. Here are some examples:

• a fir tree, in which each branch is approximately like the whole tree and each sprig is like its branch;

• the geomorphological profile of the mountains and clouds;

• the coastal strips, characterized by a profile made up of protrusions and recesses;

• ice crystals that have a variety of highly symmetrical shapes:

  Photo by Wilson Bentley.

• some leaves, like those of a fern:

  Photo of a real fern Computer-generated image of a fern by self-similarity.

• some vegetables, such as Romanesco cauliflower:

  In this type of cauliflower each head, as well as being a faithful copy of the whole vegetable, appears indistinguishable from it if observed by a magnifying glass. However, even if each sprout is composed of a series of smaller ones, the model never reaches infinitesimal dimensions.

FRACTALS, SPIRALS AND GOLDEN NUMBER

The basic element of the fractals are spirals. A particular case is represented by the logarithmic spiral which is considered a fractal, due to its self-similarity property.

Among the most beautiful images of fractals there are some whose shape recalls that of the golden spiral. An example is given by the famous Mandelbrot set, which, inside, has the Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21. Therefore, it can be said that the Mandelbrot spiral derives from the golden spiral, which can be obtained, with good approximation, using the numbers of the Fibonacci sequence. The connection between the Fibonacci sequence and the golden number is given by the ratio between two consecutive terms of the sequence; in fact, this ratio tends to approximate, better and better, the golden number φ = 1,618033.... gradually proceeding in the series of positive integers belonging to the sequence.

Detail of the Mandelbrot spiral Detail of an island.

Another example of the connection between the concepts of fractal, Fibonacci sequence and golden number is the Romanesco cauliflower. In fact, the fractal growth of the shoots of this vegetable occurs according to a logarithmic spiral; moreover, the number of spirals on the head of the vegetable corresponds precisely to a Fibonacci number (8,13):