One important feature of the mammalian immune system is the highly specific binding of antigens to antibodies. Antibodies generated in response to one infection may also provide some level of cross immunity to other infections. One model to describe this cross immunity is the notion of antigenic space, which assigns each antibody and each virus a point in Rn. Past studies using hemagglutination data have suggested the dimensionality of antigenic space, n, is low. We propose that influenza evolution may be modeled as a Gaussian random walk. We then show that hemagluttination data would be consistent with a walk in very high dimensions. The discrepancy between our result and prior studies is due to the fact that random walks can appear low dimensional according to a variety of analyses including principal component analysis (PCA) and multidimensional scaling (MDS). A high dimensionality of antigenic space is of importance to modelers, as it suggests a smaller role for pre-existing immunity within the host population.