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We live in an expanding three-dimensional universe. But what is the shape of the universe? As astronomers try to answer this question empirically, mathematicians are visualizing the possibilities. Recent predictions from cosmology, say Adams and Shapiro, indicate that the universe is most likely one of 10 of the shapes that topologists call Euclidean manifolds. They look closely at these shapes, beginning by visualizing two-dimensional analogues. The possibilities are startling: Imagine seeing copies of yourself in front and behind, yet each with a quarter-twist. Such a "quarter-twist cube space" is but one possible explanation for current observations, which astronomers hope to refine over the next decade by finely mapping and analyzing cosmic microwave background radiation, which allows us to effectively look at past universes.