In an interview with Correio Braziliense, a researcher explains the mathematics behind art
What distinguishes a work of art created by a human artist from an image produced by artificial intelligence? A study suggests that the answer may lie in mathematics. The study was the subject of an article published by Correio Braziliense on June 7, which featured IMPA researcher Jethro Van Ekeren explaining the mathematical concepts behind the discovery.
Titled “The ‘hidden’ mathematics in abstract art”The article presents the results of a study conducted by researchers at the University of Warsaw in Poland and the University of Hertfordshire in the United Kingdom. Published in the journal PLOS Computational Biology, the study identified a structural pattern shared by abstract paintings produced by renowned artists, but absent in images generated by artificial intelligence.
To arrive at this result, the scientists used a mathematical tool known as persistent homology. By Correio Braziliense, Van Ekeren explained that the method allows visual elements to be transformed into data that can be compared quantitatively.
“This tool is used to quantitatively characterize certain aspects of an object’s shape. It is divided into a hierarchy (H0, H1, H2, etc.), with each level measuring a different characteristic,” he said.
According to the researcher, in persistent homology, the object of study is the image itself. “Or, more precisely, the darker regions of an image,” he explained to the reporter.
This technique allows us to observe how shapes and contours emerge, disappear, and transform across different layers of the image, producing mathematical representations capable of revealing patterns invisible to the human eye.
The study analyzed works by artists such as Wassily Kandinsky, Mark Rothko, Jackson Pollock, Kazimir Malevich, and Maria Jarema, as well as works by the Polish painter Lidia Kot. It then compared these paintings with images generated by artificial intelligence systems trained to reproduce abstract aesthetics.
One of the study’s key findings concerns what is known as Alexander’s duality, a concept in topology that describes the relationship between structures found at the edges and within an image. By measuring the degree to which this mathematical symmetry is broken, the researchers observed that works produced by human artists exhibited virtually the same proportion of rule violations, around 0.4.
In the article, Van Ekeren explained that Alexander’s duality corresponds to an exact mathematical symmetry under idealized conditions, but that it undergoes slight changes when applied to real images.
“This is a perfect symmetry, but it holds true strictly only in an idealized situation: an infinitely large, borderless image. In real images, the interactions between the structures present in the image and its edge—not to mention the noise introduced by the digitization process—produce small deviations from the exact duality,” he said.
According to the study’s authors, the results suggest that different artists, even without explicit knowledge of the mathematics involved, tend to organize shapes and colors in similar ways when creating their compositions. In contrast, images generated by artificial intelligence did not exhibit the same regularity observed in human-created works.