Functional activity of the human brain mainly occurs on the brain surface in the thin sheet called the grey matter. This surface is highly convoluted with many folds and fissures that vary in shape and location from person to person. The complexity of the brain surface and the individual variability makes it difficult to localize and compare functional activity between individuals. "Unfolding" and flattening the cortical surface can assist in identifying functional foci obtained from PET and functional MRI data.
It is impossible to flatten a surface embedded in 3-space without introducing areal and linear distortion. However, the Riemann Mapping Theorem states that conformal (angle-preserving) maps exist. I will discuss a method that I am using to obtain an initial approximation of the conformal (flat) map of the human brain. These maps are produced using a novel computer realization of the Riemann Mapping Theorem that finds a circle packing (a collection of tangent circles) of a triangulated surface. These maps exhibit conformal behavior in that angular distortion is controlled and they can be created in the Euclidean and hyperbolic planes and on a sphere. Some of the topological and computational aspects of producing these maps will be discussed and quasi-conformal maps of the human cerebrum and cerebellar surface will be presented.