Art is subjective. No algorithmic index extracts the parameters for beauty. Yet, quantifiable clues to an artwork’s creation could help track style, technique, or artist evolution. The standard fractal analysis, which uncovers how a pattern propagates at smaller and smaller scales, is a popular measure but is often misleadingly simplistic and unreliable.

McDonough and Herczyński modified this technique to produce a unique and highly sensitive function, called a fractal contour.

“The idea that art can have fractal properties has been around for decades,” said author Andrzej Herczyński. “However, many researchers report a single fractal dimension for a painting. Imagine a Jackson Pollock painting several meters long and a few meters high with intricate, multilayered, and varying patterns. Assigning just one index reduces all this complexity to a single number.”

In addition, the values of the fractal dimension are often inconsistent because the grid used in this analysis is arbitrary. To remedy this, the team averaged the values returned for every possible grid position at each scale and carefully considered the effects of the boundaries of the artwork.

After running their code, the team analyzed the resulting scaling plot to obtain a function – the fractal contour. They tested this approach on works by Jackson Pollock, Piet Mondrian, and Frank Stella to understand what the function reveals about abstract art.

“We are only beginning to understand all the information encoded in these contours,” said Herczyński. “The positions of the minima are related to how close the pattern’s elements are to each other, and denser patterns result in shallower contours, but much remains to be discovered. We will continue to explore this technique, and we invite others to as well.”

Source: “Fractal contours: Order, chaos, and art,” by John McDonough and Andrzej Herczyński, Chaos (2024). The article can be accessed at https://doi.org/10.1063/5.0207823.

This paper is part of the Topics in Nonlinear Science: Dedicated to David K. Campbell’s 80th Birthday Collection, learn more here.