This paper presents a novel statistical framework for human cortical folding pattern analysis that relies on a rich multivariate descriptor of folding patterns in a region of interest (ROI). The ROI-based approach avoids problems faced by spatial-normalization-based approaches stemming from the deficiency of homologous features between typical human cerebral cortices. Unlike typical ROI-based methods that summarize folding by a single number, the proposed descriptor unifies multiple characteristics of surface geometry in a high-dimensional space (hundreds/thousands of dimensions). In this way, the proposed framework couples the reliability of ROI-based analysis with the richness of the novel cortical folding pattern descriptor. This paper presents new mathematical insights into the relationship of cortical complexity with intra-cranial volume (ICV). It shows that conventional complexity descriptors implicitly handle ICV Differences in Different ways, thereby lending Different meanings to “complexity”. The paper proposes a new application of a non-parametric permutation-based approach for rigorous statistical hypothesis testing with multivariate cortical descriptors. The paper presents two cross-sectional studies applying the proposed framework to study folding Differences between genders and in neonates with complex congenital heart disease. Both studies lead to novel interesting results.