Over the last several years my collaborators and I have evaluated how children, adults, and people with focal brain injury (stroke) update perceptual representations to morphing, ambiguous figures and their representations of the statistical structure of their environment probed with a variety of "games" played against a computer. In the course of this work we have seen several examples where the dynamics of participant responses point to an underlying geometrical structure. Drawing on the methods demonstrated in the first week of the workshop, I will highlight particular examples from our behavioral data and ask how the newer geometrical methods might help us better delineate and understand these patterns. In addition, I will also ask whether these approaches and others such as applied category theory, might be used to tell us "why" we see what we see; is there a way to infer a cognitive, conceptual space from the structures of a posited particular category and behavioral patterns of human choice?