Geometry and Topology in Mainstream Culture, Part III

几何游戏软件（将在本项目中开发）采用了现代几何和拓扑学的一些最基本的思想-即多连通空间，对称性和更高的维度-并将其带给学生，艺术家和其他外行的广泛受众。 传统上，数学教学要求很高：数学家经常要求学生按照数学家自己的条件来满足他们。 目前的项目扭转了这种安排，在他们熟悉的环境中，如游戏或艺术品中，以他们自己的方式将美丽的数学思想带给公众。 为了使每个空间或对称图案栩栩如生，该软件创造了一个真实的物理对象的错觉，人们可以把它握在手掌中玩。 在玩游戏或画画的同时，人们建立了对数学的直观理解。 这个项目的主要目标是让非专业人士享受现代几何，就像非专业人士享受文学和音乐一样;同时希望这种接触“先进”的几何思想可以帮助培养和激励下一代数学家，科学家和工程师。就具体的数学内容和软件而言，目前的项目有三个目标。 第一个目标是改进和扩展现有的五个几何游戏旗舰应用程序：的环面游戏（八个熟悉的游戏，但在平面2流形和3流形上玩），KaleidoPaint（在17个欧几里德壁纸组的作用下创作出五彩缤纷的绘画作品），KaleidoTile（探索球面、平面和双曲面上的多面体和镶嵌），曲面空间（球形、平面和双曲三维流形的飞行模拟器）和4D Draw（四维矢量图形）。 第二个目标是将Hyperbolic Games（双曲2流形上的常见游戏）从一个小应用程序提升到旗舰质量。 第三个目标是开发两个全新的应用程序：4D迷宫（四维空间中的迷宫，作为学习可视化4D的辅助工具）和Cross Chase（投影平面游戏）。 在硬件方面，鉴于人们普遍使用智能手机和平板电脑，为Android和iOS提供所有这些应用程序的版本是一个特别高的优先事项。 当然，Macintosh和Windows版本也将开发。

The Geometry Games software (to be developed in this project) takes some of the most fundamental ideas of modern geometry and topology -- namely multi-connected spaces, symmetries and higher dimensions -- and brings them to a broad audience of students, artists, and other laypeople. Mathematics instruction has traditionally been very demanding: mathematicians often ask their students to meet them on the mathematicians' own terms. The current project turns this arrangement around, bringing beautiful mathematical ideas to the general public on their own terms, in a context they are familiar with, like games or artwork. To bring each space or symmetry pattern to life, the software creates the illusion of a real physical object that people can hold in the palm of their hands and play with. While having fun playing a game or painting a picture, people build up an intuitive understanding of the mathematics. This project's primary goal is to let nonspecialists enjoy modern geometry in the same way that nonspecialists may enjoy literature and music; at the same time the hope is that this exposure to "advanced" geometrical ideas may help prepare and inspire the next generation of mathematicians, scientists, and engineers.In terms of specific mathematical content and the software that brings it to life, the current project has three goals. The first goal is to improve and extend the five existing Geometry Games flagship apps: Torus Games (eight familiar games, but played on flat 2-manifolds and 3-manifolds), KaleidoPaint (create colorful paintings under the action of the 17 Euclidean wallpaper groups), KaleidoTile (explore polyhedra and tilings on spherical, flat and hyperbolic surfaces), Curved Spaces (flight simulator for spherical, flat and hyperbolic 3-manifolds) and 4D Draw (4-dimensional vector graphics). The second goal is to raise the Hyperbolic Games (familiar games on hyperbolic 2-manifolds) from a minor app to flagship quality. The third goal is to develop two completely new apps: 4D Maze (mazes in 4-dimensional space, as an aid to learning to visualize 4D) and Cross Chase (a projective plane game). In terms of hardware, delivering versions of all these apps for Android and iOS is an especially high priority, given people's ubiquitous use of smartphones and tablets. Versions for Macintosh and Windows will of course also be developed.