The Dodgson winner seems very intuitive and reasonable: when a Condorcet winner doesn't exist, pick the candidate that is closest, under some measure, to being a Condorcet winner. However, Dodgson's method is computationally intensive. Approximate methods are more tractable. By placing these methods in a geometric framework, we can understand how different approximations to the Dodgson winner might identify different candidates as the winner. Furthermore, our framework provides intuition concerning some unexpected properties of Dodgson's method.