It is impossible to flatten a surface embedded in 3-space without introducing linear and areal distortion. However, the Riemann Mapping Theorem states that conformal (angle-preserving) maps exist. I will present a novel computer realization of the Riemann Mapping Theorem that uses circle packings to compute an initial approximation of the conformal (flat) map of a tessellated surface. I will also present results obtained by applying this method to data from the human brain. A tessellated surface representing the grey matter of the human brain can be constructed from magnetic resonance (MR) images. Because of its complexities, there is great interest by the neuroscience community to "unfold" and flatten this surface to create a flat map of the brain. These flattened surfaces from different subjects can then be compared to elucidate information about individual differences in the functional organization of the brain.