Mathematics and the art of fractals

Mathematics and the art of fractals by Luca pinto

Today I have the pleasure of hosting a post written by Eng. Luca Pinto, who kindly offered because it is my partner and of course as you can imagine he had no alternative …. However, he draws a connection between fractals and art. I love fractals, or rather their representation, they are beautiful, the formula will not speak. I like fractals because they are themselves all over, from any point of view. Also to divide them, torchiarli, chop, shred, reproduce themselves. And how can we think of the artist? Between poetry, utopia, obstacles, judgments, the artist always offers a picture of himself ‘consistently, in every aspect of his existence, his life and work. So I would say with enthusiasm to Ulysses: the fractal’s me.

Good scholarly reading.

The man at some point, they are no longer content with the representation of the ideal world, founded by Plato and perfected by thinkers such as Newton and Kant. The world of ideas, the noumenon, the “linear” systems were no longer sufficient. It was time to find a new path, to the representation of reality.In mathematics, thanks to Gauss early ‘800, complex numbers find their final location. Complex numbers are represented as A + iB. Complex numbers are equipped with a “real” part (A) and of a part “imaginary” (iB, where the means “imaginary”). The “imaginary” part was created to represent the impossible: the number which when multiplied by itself, -1! … Try that with all the numbers, from the simplest to the famous greek Pi and will not succeed!In 1975 a brilliant employed jew IBM Polish, Mendelbrot, use complex numbers to explain some as the “dynamic” systems change over time. Mendelbrot invents “fractals.” Fractal means “divided by”, because the fractal does not have a “full” size. As if to say that it is necessary to look at the reality “break” the veil and use an “imaginary” (complex numbers!) And choose a “fractional” dimension … a “dimension between the size” system that ranks among the line ( size n = 1), the square (size n = 2) and the cube (size n = 3).

The fractal is evidence of contact between mathematics and art: fractal exists in the ability to “imagine” new solutions, how does the artist when he looks inside himself and creates his work. Fractal exists in the ability to “travel between dimensions,” how does the artist, who “opens” a window in the chart and breaks the rules imposed by preconceptions.

Fractal equations are disarmingly simple. The whole equation Mendelbrot, if written on a blackboard is far from the image of mathematics that many films convey to us … blackboards filled with complex formulas and graphic signs curious. The beauty of fractals is so important … and free from any false footprint as a sumi-e Japanese or a work of Fontana.

Yet the fractal is of unfathomable complexity. If we zoom in the graphic representation of a fractal is always found the same form. As with a DNA fractal repeats one’s self, no matter how we break it. A foundational feature of fractal is the dilation: remains unaltered and intact regardless of the magnification. The fractal includes everything in Himself: it can not be separated into parts and analyzed. In every part is the whole. This fractal is like art: it contains a seed of life that makes no sense chop. Cut into a piece of art pieces, analyze the detail will not allow us to own the process: how a fractal is to be taken as a whole, complex and integral.

Fractals are not an invention of mind: as the art is a living reality. Exist in Romanesco cabbage, that my companion aptly called “fractal hell” (… and imagine how it responds the greengrocer to our request when we go to the grocery store!), In the structure of the alveoli in the lungs, in the form of snowflakes, in motion of particles in a glass of water, the movement of the flames in the air. Here is another point of contact between fractals and art: both speak of the living and the world of reality.