Abstract

This article describes and demonstrates the basic phenomenon of fractals that appear either in nature or in art such as in many of Escher's artwork. Fractals may be encountered inside human's body as demonstrated by the artist Grey or in the external surroundings, such as in trees painted by Friedrich. Fractals are a class in the mathematics of complex geometric shapes that commonly exhibit the property of self-similarity, such that a small portion of it can be viewed as a reduced scale replica of the whole. The term fractal is derived from the Latin word fractus ("fragmented," or "broken"). Fractals are distinct from the simple figures of the classical Euclidean, geometry (the square, the triangle, the circle, the sphere, etc.) established about 2300 years ago. They are capable of describing the many irregularly shaped objects or spatially non-uniform phenomena in nature that cannot be accommodated by Euclidean geometry. Although fractals exist in nature almost everywhere, they were defined and developed mathematically by the Scientist Benoit Mandelbrot, the "father" of fractals, only in the seventies of the 20th century. Mandelbrot, born in Poland in 1924, was living in France where he did most of his scientific work.

As elaborated in the article, artificial fractals are obtained by the solution of non-linear mathematical equations and presentation of the results in a graphical form. The patterns created in this way are extremely aesthetic, creating, thus, of a new art, the "fractals art", by the sciences of mathematics.

What are fractals?

Fractals are defined as geometrical patterns that reproduce themselves on any scale. This means that as viewers peer deeper into the fractal image, they notice that the shapes seen on one scale are similar to the shapes seen in the detail in another scale. Thus, fractals major characteristic is repeatability of a certain shape, which may be detected by “zooming in”. Moreover, fractals enable one to see the infinity by continuous zooming in into the repeating images. Fig.1 demonstrates the definition in the following way. On the right-hand side one sees a single hand, which is the basic pattern. By “zooming in” at each finger, the basic element appears again as seen in the central picture. By “zooming in” again, the above phenomenon repeats itself again as seen on the left-hand side and again and again.

Fig.1. Demonstration of fractals and the “zooming in” effect

Not too many of us are familiar with the fact that fractals is a fundamental phenomenon in our universe almost from its beginning when God created trees, stars, Adam and Eve, etc. Although, at first glance fractals demonstrate a chaotic order, their repetition again and again on any scale indicates an astounding order of the creation. Fractal patterns, demonstrated in the following, appear in our body, in trees, in sea waves, in clouds, in microbes and in lightning. Spaceship photographs of earth demonstrate also its fractal pattern. On the other hand, from earth, space shows also a fractal pattern, namely, spherical stars which repeat themselves by “zooming in” more and more into space by telescopes. Newton’s General Law of Gravitation may explain the order in the distribution. It governs the balance between the forces according to the mass of the different stars and the distance between them.

As elaborated in the following, artificial fractals are obtained from a solution of mathematical equations and presentation of the results in a certain way. The patterns formed in this way are extremely aesthetic, namely, the new art of fractals is generated via the mathematical sciences.

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