The mammalian skull has been studied as several separate functional components for decades, but the study of modularity is a more recent, integrative approach toward quantitative examination of independent subsets of highly correlated traits, or modules. Although most studies of modularity focus on developmental and genetic systems, phenotypic modules have been noted in many diverse morphological structures. However, few studies have provided empirical data for comparing modules across higher taxonomic levels, limiting the ability to assess the broader evolutionary significance of modularity. This study uses 18-32 three-dimensional cranial landmarks to analyze phenotypic modularity in 106 mammalian species and demonstrates that cranial modularity is generally conserved in the evolution of therian mammals (marsupials and placentals) but differs between therians and monotremes, the two extant subclasses of Mammalia. Within therians, cluster analyses identify six distinct modules, but only three modules display significant integration in all species. Monotremes display only two highly integrated modules. Specific hypotheses of functional and developmental influences on cranial bones were tested. Theoretical correlation matrices for bones were constructed on the basis of shared function, tissue origin, or mode of ossification, and all three of these models are significantly correlated with observed correlation matrices for the mammalian cranium.

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There are many scientific and engineering applications where an automatic detection of shape dimension from sample data is necessary. Topological dimensions of shapes constitute an important global feature of them. We present a Voronoi based dimension detection algorithm that assigns a dimension to a sample point which is the topological dimension of the manifold it belongs to, Based on this dimension detection, we also present an algorithm to approximate shapes of arbitrary dimension from their samples. Our empirical results with data sets in three dimensions support our theory.

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