Geometry and Topology in Mainstream Culture

几何拓扑学曾经是一个边缘话题，在研究生院研究得很好，但在主流文化中几乎不为人知。令人高兴的是，近年来公众的兴趣有所增加。有效地向广大读者展示几何拓扑的关键是要记住，仅使用文字的解释肯定是失败的。使用图片的解释会传达出一种模糊的意图感。但要深入理解多连通空间的拓扑结构，学习者必须亲自体验空间。其中一个私家侦探本项目的主题是提供计算机软件，使儿童和其他非专业人士能够直接体验2-流形、3-流形和其他结构。例如，私家侦探现有的环面游戏已经被证明对五年级以上的学生有效，引导他们思考多连接空间的延伸概念（特别是平面的2环面和平面克莱因瓶）。目前的项目旨在保持这种儿童友好的方法，同时扩展软件的数学范围，包括球面和双曲2流形以及3流形。虽然数学家和物理学家对空间的理解已经取得了巨大的进步，但普通公众的理解仍然落后。部分原因是，虽然现代空间概念本身并不难理解，但用语言来解释却极其困难。用图片解释效果会好一点，但在实践中，它们通常也被证明是不够的。对于学生（或非专业的成年人）来说，唯一真正有效的方法是让学生自己去体验新的空间。侦探现有的Torus Games软件可以让学生对有限但没有边界的空间产生直觉性的理解；该软件广泛应用于初中和高中，对五年级以上的学生有效。目前的项目旨在保持这种儿童友好的方法，同时扩展软件的数学范围，包括弯曲空间和平面空间，以及三维空间和“玩具”二维空间。宇宙学家目前正在研究一个假设，即真实的宇宙可能是有限的，但无论这个假设最终是被接受还是被拒绝，各种各样的空间（平面和弯曲，有限和无限，二维和三维）在物理学和数学中发挥着无处不在的作用，值得我们在整个文化中得到更广泛的理解。

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.