It wasn’t a happy line that of Nikita Khrushchev, who called «painted by the tail of a donkey» the works of Jackson Pollock. It was a stupid thing that arose from the obscure and abstruse meanderings of Soviet orthodoxy, according to which abstract art was a bin, a plaything of capitalism. So much so that the Muscovite Kandinsky, who had inaugurated it in the ’10s of the twentieth century, had to escape to Paris, where he died as a French citizen.

For sure, abstract painting raises interpretation issues. If the subject is reflected in a superficial way, it seems logical that people like portraits, landscapes and action scenes better than scribbled color patterns on a canvas with no bearing on natural reality.

Pollock’s paintings, then, are even more enigmatic than those of Kandinsky, where reassuring geometric or even parafigurative shapes prevail: Pollock’s really look as randomly made, from the tail of the donkey! Yet they have a huge success and I myself, while not an art expert, go crazy for them.

The truth is that geometric shapes exert a charm on us, and maybe even the random or almost random ones do: just think of waves or clouds. Ever since Plato’s time we reason about why we humans can feel good before a work of art. Over the past 50 years, with the advent of computers and ever more refined tools of scientific investigation, such as optical scanners or magnetic resonances showing the functional centers of pleasure in action in people’s heads, research has taken it quite seriously.

Already in high school I was struck by a collection of essays edited by Umberto Eco: Aesthetics and Information Theory (Estetica e teoria dell’informazione, Bompiani 1972), in which the tools of Shannon’s theory and related works were used to measure and explain the meaning of artistic perception. There also have been furious fads, such as when people wanted to recognize the golden ratio throughout, from Egyptian Pyramids to Raphael to sports cars: in his The equation that could not be solved (L’equazione impossibile, BUR 2005), Mario Livio dismantled several of those speculations, at the same time telling us the importance of symmetry in our sensory perception.

Now, since twenty years but exponentially growing, fractal aesthetics has established itself, and with it the search for fractality in abstract art.

Fractal aesthetics

Fractal geometry, namely the one made of everywhere continuous but nowhere differentiable curves, seems more suitable to describe the real world than the ordinary geometry of idealized regular shapes. After all, we have never met a circle or a triangle: only mild approximations. And since fractal geometry has been invoked to analyze chaotic phenomena and explain the shape of the “phase space” of complex systems, it seems reasonable to also apply it to the tail of the donkey daubing a canvas.

According to University of Oregon’s physicist Richard Taylor, Jackson Pollock’s painting is made of fractal patterns. This is indeed the first case ever discovered and studied of a fractal generated by a human being, that is not found in nature or generated by a computer (hereinafter, there was talk of fractals concerning paintings by Leonardo, the Eiffel Tower and other works of art).

Taylor et al. measured the entire production of the painter and found a fractal dimension increasing over time: from 1.3 of 1945 to 1.9 in 1950. (The dimensions of the fractal world are comprised between 1 and 2. Differentiable curves of the ordinary geometry have a dimension of 1. The Peano curve, which can fill the two-dimensional space, has dimension 2. A piece of paper crumpled up and thrown in the trash bin has D = 1.5).

In parallel to the studies on Pollock, and even before that, a thriving research had been born on the alleged aesthetic of fractals. How do humans perceive fractals? Which do we like most and which least? (Here goes again the theme of symmetry and its charm). In the mid-’90s, several measurements were made ​​using human observers and software-generated shapes, but failed to find any correlation between the fractal dimension D and the pleasantness of the sensations experienced by human recipients.

Stubborn Taylor did not give up, and in 2002-2003, enlisting a bevy of staff psychologists, showed to 200 human guinea pigs three types of fractals: those found in nature (clouds, trees, cauliflower, etc.), those generated by the software … and the paintings of Pollock. He believes to have found a prevailing aesthetic preference for fractal dimensions between 1.3 and 1.5.

Fractals as a bridge between science and art

Following these (controversial) studies, fractal art movements have started to appear. These authors, unlike Pollock and other artists predating Mandelbrot’s works in the 70’s, have the explicit purpose of painting fractally and/or are consciously influenced by fractal geometry -even though in reality they rarely understand it well. They, for example, often confuse random and fractal shapes.

Another confusion concerns the relationship between mathematics and art. Today it is common to hear that fractal geometry be the definitive trait d’union between the artistic and the scientific worlds.

Yet the link between mathematics and art is more complex, and far more ancient than the discovery of fractals (a rationalization, due to Mandelbrot, of work by Weierstrass, Cantor, Koch, Poincaré, Klein, Julia and other early twentieth century scholars): just think of symmetry, Keats’ «truth is beauty», or the eternal talks on the beauty of mathematics, occurring long before Mandelbrot and Pollock.

Although today we find them specially intriguing, fractals are but one of the links between science and art, not the only or even the main one. They are one of the arches of a complex bridge.

The role of the observer

I admire Taylor’s and similar works and I believe they are of great interest. It is important to understand how aesthetic perception in humans si formed.

But at times these studies have us forget that art perception is something more complicated than a beautiful sequence of numbers and a chi-square test. Aesthetics is not merely a function of the work of art but also, and strongly, of the observer’s cultural level.

Therefore, no matter how many tricks we devise to try to understand what a man or a chimp feels when looking at a landscape or a symmetrical shape or a fractal, we are still far from an holistic explanation of aesthetic perception. We should organize experiments that filter out the “noise” of culture. With monkeys, it can be done: with humans it’s more difficult. Even artists that are great but popular, and that everyone likes, like Fellini, Caravaggio, van Gogh, Haring or Verdi, are perceived differently by me and by the historians of their respective arts.

It is a problem similar to that which arises in the measurement of IQ. Many of the tests contain a cultural component, which interferes with the “innate” intelligence background. Subjects who master the language better and/or are more familiar with the cultural references made in the tests, appear more intelligent even though perhaps they are not.

Paolo Magrassi Creative Commons Non-Commercial – No Derivative Works
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