In this paper, using certain conformal mappings from complex function theory, we give an explicit method for flattening the brain surface in a way which is bijective and which preserves angles. The conformal equivalence arises as the solution of a certain elliptic equation on the surface. Then from a triangulated surface representation of the cortex, we indicate how the procedure may be implemented using finite elements. Further, we show how the geometry of the cortical surface and gray/white matter boundary may be studied using this approach. Hence the mapping can be used to obtain an atlas of the brain surface in a natural manner.