Invariance and Meaningfulness in Phenotype spaces

Mathematical spaces are widely used in the sciences for representing quantitative and qualitative relations between objects or individuals. Phenotype spaces-spaces whose elements represent phenotypes-are frequently applied in morphometrics, evolutionary quantitative genetics, and systematics. In many applications, several quantitative measurements are taken as the orthogonal axes of a Euclidean vector space. We show that incommensurable units, geometric dependencies between measurements, and arbitrary spacing of measurements do not warrant a Euclidean geometry for phenotype spaces. Instead, we propose that most phenotype spaces have an affine structure. This has profound consequences for the meaningfulness of biological statements derived from a phenotype space, as they should be invariant relative to the transformations determining the structure of the phenotype space. Meaningful geometric relations in an affine space are incidence, linearity, parallel lines, distances along parallel lines, intermediacy, and ratios of volumes. Biological hypotheses should be phrased and tested in terms of these fundamental geometries, whereas the interpretation of angles and of phenotypic distances in different directions should be avoided. We present meaningful notions of phenotypic variance and other statistics for an affine phenotype space. Furthermore, we connect our findings to standard examples of morphospaces such as Raup's space of coiled shells and Kendall's shape space.