Phalangeal formula, topological insights, and symmetry in the manus of Lepidosauria

The phalangeal formula (PF) describes the skeletal configuration of the digital regions of the autopodia. The plesiomorphic phalangeal formula (PPF) for the Lepidosauria manus is 2-3-4-5-3. Previous research identified 53 unique PFs in lepidosaurians. Here, we propose a new analytical method treating the PF as a matrix, applying image processing techniques to analyze new organizational patterns including symmetry. We quantified the degree of symmetry of the 53 lepidosaurian PFs through a symmetry score capturing the overlap between the original images and their transformed counterparts after specific geometric operations. We found that the PPF revealed a previously unnoticed diagonal axis of symmetry, a pattern highly improbable compared to thousands of randomly generated PFs. In a PCA of the 53 lepidosaurian PFs symmetry scores, PC1 was significantly correlated with the total number of phalanges whereas PC2 reflected the variability in phalangeal counts among different digits. The PCA effectively summarized the underlying variance of the symmetry scores data, elucidating a reduction trend in phalanx count (PC1) and the degree of diversity of PF arrangements (PC2). This approach is useful for addressing symmetry in meristic characters and systems of articulated pieces which could reveal hidden symmetry patterns in other lineages.