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Across the nervous system, neurons often encode circular stimuli using tuning curves that are not sine or cosine functions, but that belong to the richer class of von Mises functions, which are periodic variants of Gaussians. For a population of neurons encoding a single circular variable with such canonical tuning curves, computing a simple population vector is the optimal read-out of the most likely stimulus. We argue that the advantages of population vector read-outs are so compelling that even the neural representation of the outside world's flat Euclidean geometry is curled up into a torus (a circle times a circle), creating the hexagonal activity patterns of mammalian grid cells. Here, the circular scale is not set a priori , so the nervous system can use multiple scales and gain fields to overcome the ambiguity inherent in periodic representations of linear variables. We review the experimental evidence for this framework and discuss its testable predictions and generalizations to more abstract grid-like neural representations. [ABSTRACT FROM AUTHOR]