If art is a math problem – basic research

By Edgard Pimentel

Newton’s binomial is as beautiful as Venus de Milo

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Mathematics inspired and favored art. Perspective, proportions, and symmetry are fundamental in the fine arts, for example. And the clove was flavorful with a good math clue. Volpi’s flags, Athos Bulcãos tiles, cubism … But what about the opposite? Does art inspire math?

On the other side of the Atlantic there is evidence of the link between art and mathematics. According to Fernando Pessoa, “Newton’s binomial is as beautiful as Venus de Milo,” but people don’t notice. Here art bestows its ideals as an arrow pointing to the beauty of the mathematical object. But maybe you can go on.

In 1954 the International Congress of Mathematicians (ICM, International Congress of Mathematicians) took place in Amsterdam. The program included an exhibition by Escher, whose work has a strongly geometric character. Just look at your finite stairs that always seem to go up. Or the coating of an airplane with a single figure (e.g. a winged goldfish) through mathematical transformations without leaving an empty space. The goldfish is a fundamental region for a group of symmetry – transformations of the goldfish that lead to themselves.

At this conference Escher had the opportunity to address scientists such as the mathematician Harold Coxeter and the Nobel Prize winner Roger Penrose, also a physicist. The correspondence with the first inspired him to finish the work of the “Grenzkreis”: the same figure replicates itself within a circle and becomes smaller and smaller as it approaches the edges.

But the opposite would also happen: the artist’s work would have at least partially motivated 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 of the work is the catalog of Escher’s exhibition from 1954. Perhaps Escher and his “Parças” are a one-way street for inspiration between art and mathematics.

On the other hand, could mathematics answer an important question in art?

Dating a work without chronological records is a relevant task for art history. Or understand if and how an artist’s style has changed over time. And math can help solve those questions. To like? Treat a painting as a mathematical object, as a function. And divide this function into smaller units. Studying these smaller units is a key that unlocks information about the artist in question.

A very efficient tool in this sense are wavelets: very special functions that, as the name suggests, look like waves, small and well-behaved. It’s extremely powerful – insofar as the JPEG format depends on you. When a painting is analyzed using wavelets, the result is a series of numbers that contain information about the painting.

In the past ten years, the Van Gogh and Kröller-Müller museums have made more than a hundred high-resolution photographs of Van Gogh’s works available for a multidisciplinary study. A group of scientists combined wavelets with machine learning and came up with surprising information. For example, they found evidence that Van Gogh’s number of brushstrokes is greater when he’s in Paris rather than Arles. One of the leading researchers in this group was the Belgian mathematician Ingrid Daubechies.

In 2018 the ICM took place in Rio de Janeiro. On that occasion, Daubechies spoke about studying Van Gogh’s works and other art problems that motivated mathematical research. Among them, the researcher spoke about the challenges of removing cracks in a painting that could reveal a text by Tomás de Aquino in a play by the Van Eyck brothers.

Art, math, and science must have much more in common than we can imagine – after all, they are ways of developing the human mind. I hope there are more and more people who realize this.

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