The Italian mathematician Giuseppe Peano (1858-1932) is best known for creating an axiom system for arithmetic that today remains the starting point for most rigorous developments of modern mathematics, but he is also famous for his construction of a curve that fills an entire planar region. This seems counterintuitive since our usual notion of a curve is that it is one-dimensional. So how could a one-dimensional object fill a two-dimensional region? Peano's curve was something of a curiosity when it appeared in an 1890 article, but a year later the German mathematician, David Hilbert (1862-1943), produced another "plane-filling" curve. At this point some mathematicians began to wring their hands and proclaim such curves to be "non-intuitive," "monstrous," and "pathological"; and they feared that these things threatened to undermine some of the most cherished concepts in mathematics. Another problem for the mathematicians, in addition to the quandary about...