Geometry and Topology in Mainstream Culture

几何拓扑学曾经是一个边缘话题，在研究生院里研究得很好，但在主流文化中几乎不为人所知。令人高兴的是，近年来，公众的兴趣有所增加。要有效地向广大受众展示几何拓扑学，关键是要记住，仅用文字解释肯定会失败。用图片解释将传达出对原意的模糊感觉。但要深入理解一个多连接空间的拓扑结构，学习者必须亲自体验这个空间。P.I.S在生活中的使命之一-也是本项目的主题-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-歧管、3-歧管和其他结构。例如，P.I.的S现有的环面游戏已经被证明对五年级以上的学生很有效，这让他们产生了多连接空间的令人兴奋的概念(具体地说，平面二环和平面克莱恩瓶子)。本项目旨在保持这种适合儿童的方法，同时将该软件的数学范围扩展到球面和双曲二维流形以及三维流形。尽管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。用图片解释的效果稍好一些，但在实践中，它们通常也被证明是不够的。对于学生(或非专业成人)来说，完全掌握新的空间概念的唯一真正有效的方法是让学生自己体验新的空间。P.I.S现有的Torus Games软件让学生们对有限但没有边界的空间有了一种直观的理解；这款软件在初中和高中广泛使用，对五年级以上的学生很有效。目前的项目旨在保持这种适合儿童的方法，同时扩展软件的数学范围，包括曲线空间和平面空间，以及三维空间和“玩具”二维空间。宇宙学家目前正在研究真实宇宙可能是有限的这一假设，但无论这一假设最终是被接受还是被拒绝，所有类型的空间(平面和曲线，有限和无限，二维和三维)在整个物理和数学中都扮演着无处不在的角色，值得我们作为一个整体在我们的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.