Multifractal luminance and aesthetic complexity in van Gogh's paintings

Vincent van Gogh's late paintings are renowned for their distinctive motion-like textures, where swirling patterns and luminous contrasts evoke the dynamics of turbulent flow. While previous studies have identified turbulence-like statistics in individual works, such as The Starry Night, a systematic quantification of visual complexity across his oeuvre has remained unexplored. Here, we apply two-dimensional multifractal detrended fluctuation analysis to four major paintings produced during the artist's final 15 months to examine how scale-dependent organization evolves across this creative period. The analysis reveals clear differences in multifractal behavior among the paintings, reflecting systematic variations in the organization of luminance fluctuations. Paintings produced during periods historically documented as psychologically unstable for the artist exhibit broader multifractal spectra and stronger deviations from monofractality, whereas those created during intervals of relative calm display more homogeneous scaling patterns. To summarize this variability, we introduce the Fractal Turbulence Index, a composite metric integrating heterogeneity and roughness, which consistently differentiates the paintings according to their degree of structural complexity. Beyond the artistic context, the study demonstrates how methods originally developed for turbulence and scale-invariant phenomena can be extended to human-generated imagery, providing a quantitative framework for understanding how order, variability, and intermittency emerge in aesthetic expression.