They can be spotted in Van Gogh’s and Pollock’s paintings, in the architecture of Angkor Wat, in the Venetian Doge’s Palace, and in Mozart’s compositions. I’m referring, of course, to fractals.
The 1990s witnessed a heated debate about Jackson Pollock’s paintings. It all began after Teri Horton purchased what was supposedly one of his paintings for $5. The authenticity of the painting was questioned by art connoisseurs – after extensive research it was decided that it was too dark in colour and painted with acrylic paints that Pollock never used. Moreover, the canvas was of standard size, which was odd, since Pollock used sail cloth, which he manually trimmed. Still, the expert opinions weren’t generally accepted. In response, mathematician Richard Taylor decided to unequivocally determine whether the painting was authentic. Using the 1948 painting Number 1 as a model, he adopted a method of box-counting: he superimposed onto canvas ‘box-shaped’ pieces at each smaller scale. Then he counted how many boxes contained the elements of the painting. The idea was to measure the extent to which the pattern covered the surface.
The number of boxes corresponds to the covered surface. Of course, this number changes depending on the size of the ‘box’. What is crucial is the rate of these changes, since it allows us to determine how many dimensions a given object has. Here’s a simple example:
The first drawing contains a one-dimensional line, which takes up three boxes. What happens when we reduce the size of the boxes by half? The line would take up six of them. The number of covered boxes is directly proportional to the first power of the scale – the drawing has one dimension. In other words, the length of the line is measured in boxes.
Now let’s look at a flat figure, which takes up two boxes. After rescaling it twice, there are eight boxes, so four times as many. Thus, the rate of change is proportional to the squared scale. The dimension equals two. In other words, the surface area is measured in boxes.
This method, as demonstrated, can be used to measure the dimension of the object, which, as we know, doesn’t have to be an integer number. Objects with a non-integer number of dimensions are characterized by self-similarity.
Pollock’s Number 1 turned out to be 1000 times self-similar, while the painting bought by Teri Horton was only 50 times self-similar. By way of comparison, a typical contour of the forest against the sky can be enlarged 250 times while preserving its self-similarity. Subsequent analysis conducted by Taylor demonstrated that the dimension of Pollock’s works corresponds to a characteristic of some parts of the natural world (for instance, the horizon line or leaf veins), i.e. around 1.3. It was also shown that Pollock developed his technique over time, since between 1943 and 1952 this number was gradually rising. What also mattered, however, was the size of the canvas and the extent to which it was covered.
It’s essential to study naturally occurring fractal patterns which – as Taylor tries to prove –impact human cognitive habits. He argues that for ages artists who intuitively drew on natural forms have relied on fractals in their creative process, since they’ve been constantly exposed to them. This way, those used to a specific dimension of an object (around 1.3) rest their eyes, which also entails a reduction in brain activity connected with image processing. This kind of rest is pleasurable and conducive to finding other analogies. As such, it fulfils a natural – according to Johan Huizinga – need of entertainment. This is also exemplified by Gestalt theory and Rudolf Arnheim’s work Art and Visual Perception. This psychophysiological research has shown that perception is governed by the simplicity principle and a resulting perceptual stability. Given that fractals are infinitely self-similar and ubiquitous in nature, it’s hard to refute Taylor’s claims. During the 1986 NASA experiment, he showed to randomly selected people four different pictures: a nature landscape, a pointillist painting depicting a tree, an abstract pattern and a blank test card. As their bodily reactions were measured, it turned out that the brain response triggered by the landscape was 44% lower than that triggered by the test card. This is why another scientist, Roger Ulrich, wanted to prove that it’s beneficial for patients recovering after surgery to stay in a room overlooking nature. Perhaps some might find it funny, but this kind of knowledge is very much needed. It’s known, for instance, that looking for a long time into the cosmic void negatively affects our psyche. Also, torture used during World War II demonstrated that sensory deprivation causes people to lose their mind very quickly.
Taylor didn’t limit his research only to that, though. He designed, for instance, a painting machine, which employs the drip technique. It can be calibrated in such a way that it creates patterns of a specified dimension, and can draw both fractal and non-fractal shapes. A subsequent study demonstrated a preference for a particular dimensional range: around 1.3.
Similar studies were also conducted in architecture and urban planning. According to the current state of research, it can be argued that already in ancient times, builders would intuitively and completely unknowingly apply a method of scaling closely corresponding to the fractal – it’s clear when we juxtapose the outside contour line of Angkor Wat and the forest-covered horizon line. Moreover, Indians tend to scale and arrange some parts of architecture, such as domes or pinnacles, according to the same irrational number proportion. The walls of Angkor War also become larger in this way. A similar pattern also characterizes Buddhist sand mandalas. In contrast, Greek and Roman architecture, which in Europe for a long time was considered a model to follow, was typically based on rational numbers. The exceptions include Venezia’s Gothic palaces: Pallazo d’Oro, Ca’ Foscari and the Doge’s Palace. Parts of cities meticulously planned around streets that spread concentrically or on single-family housing estates also form patterns similar to fractals. One fine example are the areas around the Elysian Fields in Paris, or Frank Lloyd Wright’s Palmer House, built for Bill Palmer in the early 1950s (the plan of the building consists of overlapping and intersecting triangles).
Another phenomenon, which I find particularly appealing, is the presence of fractals in the works of pioneering expressionists and abstractionists. It turns out that Benoît Mandelbrot admired the structure of František Kupka’s paintings. Kupka, a Czech artist unrecognized by his contemporaries, was one of the pioneers of abstractionism, who created large canvas paintings saturated with his favourite music. Fascinated with Bach’s counterpoint rule, he transposed it into his works through the use of contrasting colours: red and sky blue. I find it surprising that the features of fractals can be identified even in these often relatively small elements. Bach’s counterpoint is a technique of combining musical lines that are independent in melodic contour yet interdependent harmonically. Their similarity corresponds to the similarity of juxtaposed forms in Kupka’s works. It seems a little surprising, however, that while fractals are evident in most of his works, they cannot be found in Chestnut Tree in Bloom (1906).
In contrast, fractals appear in all of Vincent van Gogh’s paintings that depict chestnuts. Van Gogh’s technique is entirely different from Kupka’s, though. As he often emphasized, his creative process was based on a close relationship with nature. In consequence, he developed a method intimately related to the presence of fractals in nature. A study conducted by a few higher education institutions in collaboration with the Kröller-Müller Museum showed that the way in which Van Gogh applied his brush strokes formed a specific pattern of repetition and that the entire structure of his works is very consistent. Although this can be seen at first glance, the study provides us with numbers that speak for themselves. After analysing the standardized scanned versions of original paintings and their copies, researchers concluded that fractal analysis can serve as a tool for detecting forgery, as well as for dividing Van Gogh’s works into periods, since each was marked by a different correlation of patterns. It was also found that the dimension of paintings he was working on just before or shortly after epileptic seizures was considerably higher than of those created when he was well. The change in form and the level of complexity of the fractal are overlapping in this case. Deeply aware of the universal quality of his source of inspiration, Van Gogh arrived at formal perfection.
Like in the case of Pollock’s paintings, fractal analysis allows us to determine when Van Gogh’s works were created. It took him only two years to considerably improve his technique, and what might appear as a mere deformation is in fact a way of striving toward perfection. This observation is based on the fact that Van Gogh was painting in a state of creative delirium, which means that he wouldn’t stop even for a second to reflect on his work. Copies, on the other hand, are always painted in a more attentive way since their conformity to the original needs to be constantly verified. To put it differently, Van Gogh developed a technique that renders a simple forgery impossible. Accusations that his style was a result of his disability are very harmful. The epilepsy he was diagnosed with and treated for made it more difficult for him to work, not easier – as some conspiracy theorists would suggest. Its only symptom is recurrent seizures during which a person loses consciousness (either completely or partly) and experiences involuntary muscles contractions. Such attacks are very exhausting, but each time Van Gogh regained strength, he would get back to work. He would often create two paintings a day and prepare a lot of drafts.
When it comes to music, which many early 20th-century painters were enthralled with, it’s worth examining the structure of sounds in a musical piece, which also frequently corresponds to fractals. It’s clear that to some extent music is self-similar by nature. This was further explored by Mozart, who did some experiments and invented maths games. He created, for instance, Musikalisches Würfelspiel, a system for using dice to randomly compose subsequent parts of a score. Changeability of sounds in time outside the structure of the fractal also implies the presence of noise, which according to many studies is natural. Various sorts of noise can be linked, for instance, to particular music genres. Luigi Russolo wrote about it in his 1913 letter to the composer Francesco Balilli Pratelli. Published in 1916 under the title The Art of Noises, it has been considered one of the most influential 20th-century theories of music. Russolo, affiliated with futurism, was both a painter and composer. In fact, at the turn of the century, there were many artists who combined visual arts and music. After all, this was a time when correspondance des arts, explored in Romanticism, was again brought into focus. Another artist who significantly contributed to the development of art theory back then was Téodor de Wyzewa. He co-created the magazine La Revue wagnérienne and his work on Wagner was really far-reaching. Wagner was recognized as a great genius, who enabled a synthesis of arts and created a true Gesamtkunstwerk.
Like in the case of paintings and musical scores, examples of architecture that reflect fractal patterns are endless. Regardless of whether a given artist was aware of such correspondences or not, they should be explored. There are some who tend to oppose innovative research methods, which seems understandable. However, each method of analysis is fine, as long as it’s applied directly to the work of art itself. Only such an approach allows us to continually expand our knowledge. This is why art and sciences should be brought together effectively.
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