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The great variability of cerebral cortical folding patterns raises major problems for the systematic study of functional-structural relationships. This paper describes a novel perspective for explaining this variability, a perspective that relies on gyri buried in the depth of the sulci. From this perspective we propose a generic model of folding, based on indivisible units, called sulcal roots, which correspond to the first folding locations during antenatal life. These units are organized according to a basic scheme allowing us to describe the cortical surface using a system of meridians and parallels. This scheme is thought to be stable across individuals at the fetal stage, and may be related to the protomap model. Variability at the adult stage is thought to result from the chaotic behavior of the folding process: inter-individual differences in cortical areas can lead to qualitatively different folding patterns. We have tested the capacity of this model to match actual cortical anatomy with a database of magnetic resonance images of 20 normal subjects, using new three-dimensional visualization tools giving access to shapes buried in the cortex.