The geometry of morphospaces: lessons from the classic Raup shell coiling model

Morphospaces are spatial depictions of morphological variation among biological forms that have become an integral part of the analytical toolkit of evolutionary biologists and palaeobiologists. Nevertheless, the term morphospace brings together a great variety of spaces with different geometries. In particular, many morphospaces lack the metric properties underlying the notions of distance and direction, which are, however, central to the analysis of morphological differences and evolutionary transitions. The problem is illustrated here with the iconic morphospace of coiled shells implemented by Raup 50 years ago. The model, which allows the description of shell coiling geometry of various invertebrate taxa, is a seminal reference in theoretical morphology and morphospace theory, but also a morphometric framework frequently used in empirical studies, particularly of ammonoids. Because of the definition of its underlying parameters, Raup's morphospace does not possess a Euclidean structure and a meaningful interpretation of the spread and spacing of taxa within it is not guaranteed. Focusing on the region of the morphospace occupied by most ammonoids, I detail a landmark-based morphospace circumventing this problem and built from the same input measurements required for the calculation of Raup's parameters. From simulations and a reanalysis of Palaeozoic ammonoid shell disparity, the properties of these morphospaces are compared and their algebraic and geometric relationships highlighted. While Raup's model remains a valuable tool for describing ammonoid shells and relating their shapes to the coiling process, it is demonstrated that quantitative analyses of morphological patterns should be carried out within the landmark-based framework. Beyond this specific case, the increasing use and diversity of morphospaces in evolutionary morphology call for caution when interpreting patterns and comparing results drawn from different types of morphospaces.