Geometry and Topology in Mainstream Culture, Part II

虽然几何拓扑学的研究人员在理解流形（多连通空间）方面取得了巨大进展，但这个概念现在才进入大众的想象。 私家侦探他目前的工作是让普通人直接体验二维流形和三维流形：每天都有数百人在iPhone和iPad上下载Torus和Klein Bottle Games，更多的人下载桌面版本，P.I.的软件出现在科学博物馆和科学频道（2011年夏季）。 本项目的目标是维护和扩展现有的软件，以便人们可以在新的设备上使用全套程序，如平板电脑和手机以及传统的台式机和笔记本电脑;编写新的软件，让非数学家以同样的用户友好的方式理解三维流形，Torus和Klein Bottle Games目前让他们理解二维流形;并开发软件以更好地理解平面镶嵌及其对称群。对有效平铺算法的实际需求也导致了一些有趣的理论进展（使用自旋空间的双曲和欧几里得类似物来管理表面的等距），这些进展将进一步进行。 在继续支持课堂使用几何游戏软件的同时，该提案同样强调将拓扑和几何思想引入共同文化。人类对几何的理解在过去两个世纪中取得了巨大进步，今天仍在继续进步。尽管如此，流行的几何概念往往仍然局限于2000年前的欧几里得几何。本项目旨在帮助公众了解最新的几何学。 主要的新观点--空间本身有一个“形状”--并不是天生难以理解，而是难以用语言解释。用图片来解释会更好一些，但在实践中，它们通常也是不够的。 对于学生（或非专业的成年人）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亲自体验新的空间。私家侦探微软现有的用于此目的的软件受到了好评（仅iOS版本每周就能获得4000+新用户）。本项目将开发和扩展该软件，让人们更好地体验三维空间以及二维空间，沿着相关的对称概念。培养公众对现代几何的理解有两个方面的动机：第一，培养下一代科学家和工程师，他们需要强大的3D可视化技能以及对几何和空间的现代理解;第二，让所有人都能理解和享受几何，作为我们共同文化的一部分，就像所有人都能享受文学和音乐一样。

While researchers in Geometric Topology have made enormous progress in understanding manifolds (multi-connected spaces), the concept is only now entering the popular imagination. The P.I.?s current work lets ordinary people experience 2- and 3-manifolds directly: Every day hundreds of people download the Torus and Klein Bottle Games for their iPhones and iPads, many more download the desktop versions, and the P.I.'s software appears in science museums and on the Science Channel (Summer 2011). The present project's goals are to maintain and extend the existing software, so people may use the full suite of programs on newer devices like tablets and phones as well as on traditional desktop and laptop computers; to write new software that will let non-mathematicians understand 3-manifolds in the same user-friendly way that the Torus and Klein Bottle Games currently let them understand 2-manifolds; and to develop software to better understand planar tilings and their symmetry groups. The practical need for efficient tiling algorithms has also led to some interesting theoretical progress (using the hyperbolic and euclidean analogs of spin space to manage the isometries of a surface) that will be pursued further. While continuing to support classroom use of the Geometry Games software, the proposal puts equal emphasis on introducing topological and geometrical ideas into the common culture.Humanity's understanding of geometry has advanced tremendously over the past two centuries, and continues to advance today. Nevertheless, the popular conception of geometry all too often remains limited to the Euclidean geometry of 2000 years ago. The present project aims to help bring the public's understanding of geometry up to date. The main new idea -- that space itself has a "shape" -- is not inherently difficult to understand, but it is 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 the new concept of space is for the student to experience the new space for him or herself. The P.I.'s existing software for this purpose has been well received (the iOS version alone gets 4000+ new users each week). The present project will develop and extend this software to let people better experience 3D spaces as well as 2D spaces, along with related concepts of symmetry. The motivation for developing the public understanding of modern geometry is two-fold: first, to prepare the next generation of scientists and engineers, who will need strong 3D visualization skills as well as a modern understanding of geometry and space; second, so that all people may understand and enjoy geometry as a part of our shared culture, just as all people may enjoy literature and music.