Fractals and fractals in nature

● What are fractals?● A fractal can be defined as

a geometric object which repeats itself in his structure in the same way on different scales. So it dosn't change his aspect even if you observe it trought a magnifying glass.

● How can we catalog them?

● The fractlas are usually grouped in different families:

● Linear fractals

● Non-linear fractals

● Random fractals

Families of fractals

● The linear fractals are those whose linear generating equation only contains terms of the first order, and therefore there is an algorithm of the linear type.

● These fractals can be studied with the aid of an imaginary doubler figures: the copier to reductions, a metaphorical machine conceived by John E. Hutchinson, a mathematician at the Australian National University in Canberra.

● Non-linear fractals are those whose generating equation has order greater than 1.

● Random fractals are built from a triangle. The midpoints of each side of the triangle are connected together and the triangle is so divided into four smaller triangles. Each midpoint is then raised or lowered by an amount chosen at random. The same procedure is applied to each of the smaller triangles, and the process is repeated endlessly.

Some historical information

● According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity in his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.

● In 1883, Georg Cantor published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals. Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called "self-inverse" fractals.

In 1975 Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set pictured in Figure 1 captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".Currently, fractal

studies are essentially exclusively computer-based.

Fractals in mathematics

● As mathematical equations, fractals are usually nowhere differentiable, which means that they cannot be measured in traditional ways. An infinite fractal curve can be perceived of as winding through space differently from an ordinary line.

How can we mesure a fractal?● The fractal dimension (or Hausdorff dimension) is a very

important parameter that determines the "degree of irregularity" of the fractal considered.

● Mandelbrot in his book titled "fractal objects" published in 1975 states that there are different methods to measure the dimension of a fractal, introduced when the mathematician tried her hand to measure the length of the coastline of Great Britain. Among these, the following:

● Is advanced along the coast a compass opening prescribed

every step he begins where the previous one ends. The aperture value h multiplied by the number of steps will provide

the approximate length L (h) of the costs, but making the opening of the compass always smaller than the numbers of steps increase, the opening will tend to zero and the length will tend to 'infinite.

Fractals in nature

● We can find many examples of fractals in nature, such as:

● The leaves of the fern

● Lightning bolt discharges inside a acrylic block or “Lichtenberg figure”

● Snowflakes

● Romanesco broccoli

A cura di:

Filippo Maria Conte