Quantifying the shape of cells, from Minkowski tensors to p-atic orders

P-atic liquid crystal theories offer new perspectives on how cells self-organize and respond to mechanical cues. Understanding and quantifying the underlying orientational orders is, therefore, essential for unraveling the physical mechanisms that govern tissue dynamics. Due to the deformability of cells this requires quantifying their shape. We introduce rigorous mathematical tools and a reliable framework for such shape analysis. Applying this to segmented cells in MDCK monolayers and computational approaches for active vertex models and multiphase field models allows to demonstrate independence of shape measures and the presence of various p-atic orders at the same time. This challenges previous findings and opens new pathways for understanding the role of orientational symmetries and p-atic liquid crystal theories in tissue mechanics and development.