04/02/2021

The mutual influence that exists between art and mathematics.

Ilustração: Linoca Souza

Reproduction from the Fundamental Science Blog, Folha de S.Paulo:

By Edgard Pimentel

Newton's binomial theorem is as beautiful as the Venus de Milo.

Mathematics has inspired and favored art. Perspective, proportion, and symmetry, for example, are fundamental in the visual arts. And the art has been well seasoned with a good dose of mathematics. Volpi's little flags, Athos Bulcão's tiles, Cubism… But what about the opposite? Does art inspire mathematics?

From across the Atlantic comes evidence of the connection between art and mathematics. According to Fernando Pessoa, "Newton's binomial theorem is as beautiful as the Venus de Milo," but people don't realize it. Here, art lends its ideals like an arrow pointing to the beauty of the mathematical object. But perhaps we can go further.

In 1954, the International Congress of Mathematicians (ICM) took place in Amsterdam. The program included an exhibition of Escher's work, which has a strongly geometric character. Just consider his finite staircases that always seem to ascend. Or the covering of a plane with a single figure (e.g., a winged fish) through mathematical transformations, leaving no empty space. The fish is a fundamental region for a symmetry group – transformations of the fish that result in the fish itself.

At that congress, Escher had the opportunity to get closer to scientists such as the mathematicians Harold Coxeter and Nobel laureate Roger Penrose, also a physicist. The exchange of letters with the former inspired him to finalize the "boundary circles" works: the same figure is replicated inside a circle, becoming smaller and smaller as it approaches the edges.

But the opposite could also be true: the artist's works may have motivated, at least in part, Roger Penrose and his father, Lionel Penrose. In a 1958 article published in The British Journal of Psychology, father and son discuss optical illusions and the perception of impossible shapes. One of the two references in the work is the catalog of Escher's 1954 exhibition. Perhaps Escher and his "partners" represent a two-way street for inspiration between art and mathematics.

On the other hand, could mathematics answer any important questions in art?

Dating a work that lacks a chronological record is a relevant task for art history. Or understanding if, and how, an artist's style changed over time. And mathematics can help unravel these questions. How? By treating a painting as a mathematical object, a function. And decomposing that function into smaller units. The study of these smaller units is a key that unlocks information about the artist in question.

One very efficient tool in this regard is wavelets: very special functions that, as the name suggests, look like tiny, well-behaved waves. And they are extremely powerful – to the point that the JPEG format depends on them. When a painting is analyzed using wavelets, the result is a set of numbers that carry information about the painting.

In the past decade, the Van Gogh and Kröller-Müller museums made available to a multidisciplinary study more than one hundred high-resolution photographs of Van Gogh's works. Combining wavelets with machine learning, a group of scientists obtained surprising information. They found evidence, for example, that the number of brushstrokes Van Gogh used is greater during the period he was in Paris than in Arles. One of the leading researchers in that group was the Belgian mathematician Ingrid Daubechies.

In 2018, the ICM took place in Rio de Janeiro. On that occasion, Daubechies discussed the study of Van Gogh's works and other problems in art that motivated mathematical research. Among them, the researcher spoke about the challenges behind removing cracks in a painting, capable of revealing a text by Thomas Aquinas in a piece by the Van Eyck brothers.

Art, mathematics, and science likely have much more in common than meets the eye – after all, they are forms of expression of the human spirit. Hopefully, more and more people will realize this.