Coordination across scales: A multifractal perspective on scientific modeling of movement

The question of how to measure movement and then model has had a history of seeming simple at first and growing more complex. Richardson found that wind velocity challenged conventional Euclidean intuitions that supported linear modeling with whole-number dimensions. Biological movement coordination is no less complex than wind, and it has proven to warrant the same non-Euclidean treatment of its form. Dexterous movement requires a nonlinear spreading out of movement degrees of freedom across multiple spatiotemporal scales. This spreading out across scales exhibits what mathematicians call "divergence," meaning that the diffusion does not tidily wrap up at any single scale. Divergence entails an excess of dimensionality beyond the Euclidean imagination, from whole-number to fractional dimensionality and from single-fractional dimensionality to multiple. This perspective examines how fractional dimensions elucidate the intricate cascade of interactions underlying dexterous human movement. Traditional models, limited by linear and discrete representations, contrast with emerging multifractal approaches that capture continuous, context-sensitive variability across multiple interacting scales. Integrating multifractal analysis and cascade dynamics of human movement variability advances a new framework for understanding action across scales, highlighting its relevance for motor control and rehabilitation.