Geometry and Topology in Mainstream Culture

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Geometric topology was once a fringe topic, well studied in graduate schools but almost unknown in mainstream culture. Happily, in recent years public interest has increased. The key to effectively presenting geometric topology to a broad audience is to keep in mind that explanations using words alone are sure to fail. Explanations using pictures will convey a vague sense of the intended meaning. But to deeply understand the topology of a multi-connected space, the learner must experience the space for him or herself. One of the P.I.'s missions in life --- and the subject of the present project --- is to provide computer software that enables children and other non-specialists to experience 2-manifolds, 3-manifolds, and other structures directly. For example, the P.I.'s existing Torus Games have already proved effective with students from fifth grade on up, leading them to the mind-stretching concept of multiconnected spaces (specifically, the flat 2-torus and the flat Klein bottle). The present project aims to maintain this kid-friendly approach while extending the software's mathematical scope to include spherical and hyperbolic 2-manifolds as well as 3-manifolds.While mathematicians' and physicists' understanding of space has made tremendous progress, the general public's understanding still lags behind. Part of the reason is that, while the modern concept of space is not inherently difficult to understand, it is extremely difficult to explain in words. Explanations with pictures work a little better, but in practice they too usually prove inadequate. The only truly effective way for a student (or nonspecialist adult) to fully grasp a new concept of space is for the student to experience the new space for him or herself. The P.I.'s existing Torus Games software lets students develop a gut-level understanding of spaces that are finite yet have no boundaries; this software is widely used in middle and high schools and is effective with students from fifth grade on up. The present project aims to maintain this kid-friendly approach while extending the software's mathematical scope to include curved spaces as well as flat ones, and 3-dimensional spaces as well as "toy" 2-dimensional ones. Cosmologists are currently investigating the hypothesis that the real universe might be finite, but no matter whether that hypothesis is ultimately accepted or rejected, spaces of all sorts (flat and curved, finite and infinite, 2-dimensional and 3-dimensional) play a pervasive role throughout physics and mathematics, and merit a broader understanding in our culture as a whole.