A modularity approach is used to study disparity rates and evolvability of sea urchins belonging to the Atelostomata superorder. For this purpose, the pentameric sea urchin architecture is partitioned into modular spatial components and the interference between modules is quantified using areas and a measurement of the regularity of the spatial partitions. This information is used to account for the variability through time (disparity) and potential for morphological variation and evolution (evolvability) in holasteroid echinoids. We obtain that regular partitions of the space produce modules with high modular integrity, whereas irregular partitions produce low modular integrity; the former ones are related with high morphological disparity (facilitation hypothesis). Our analysis also suggests that a pentameric body plan with low regularity rates in Atelostomata reflects a stronger modular integration among modules than within modules, which could favors bilaterality against radial symmetry. Our approach constitutes a theoretical platform to define and quantify spatial organization in partitions of the space that can be related to modules in a morphological analysis.

An eutactic star, in a n-dimensional space, is a set of N vectors which can be viewed as the projection of N orthogonal vectors in a N -dimensional space. By adequately associating a star of vectors to a particular sea urchin we propose that a measure of the eutacticity of the star constitutes a measure of the regularity of the sea urchin.Then we study changes of regularity (eutacticity) in a macroevolutive and taxonomic level of sea urchins belonging to the Echinoidea Class. An analysis considering changes through geological time suggests a high degree of regularity in the shape of these organisms through their evolution. Rare deviations from regularity measured in Holasteroida order are discussed.

A basic pattern in the body plan architecture of many animals, plants and some molecular and cellular systems is five-part units. This pattern has been understood as a result of genetic blueprints in development and as a widely conserved evolutionary character. Despite some efforts, a definitive explanation of the abundance of pentagonal symmetry at so many levels of complexity is still missing. Based on both, a computational platform and a statistical spatial organization argument, we show that five-fold morphology is substantially different from other abundant symmetries like three-fold, four-fold and six-fold symmetries in terms of spatial interacting elements. We develop a measuring system to determine levels of spatial organization in 2D polygons (homogeneous or heterogeneous partition of defined areas) based on principles of regularity in a morphospace. We found that spatial organization of five-fold symmetry is statistically higher than all other symmetries studied here (3 to 10-fold symmetries) in terms of spatial homogeneity. The significance of our findings is based on the statistical constancy of geometrical constraints derived from spatial organization of shapes, beyond the material or complexity level of the many different systems where pentagonal symmetry occurs.