The cortical surface of the brain is very convoluted, with many folds and fissures. The cortical surface is topologically equivalent to a sheet, so it is possible to "unfold" it and create a cortical flat map of the brain, which may reveal functional and anatomical spatial relationships that were not previously apparent. It is impossible to flatten a surface with intrinsic curvature (such as the brain) without introducing linear and areal distortion, but a 150-year old mathematics theorem proves that it is possible to preserve angular (conformal) information under flattening. I will describe a novel computational method which applies the mathematical theory of circle packings to MRI scans to create discrete conformal flat maps. I will present flat maps of the cortical surface obtained using this approach and describe how I am using these maps to investigate depression in twins.