Geometry and Topology in Mainstream Culture, Part III

主流文化中的几何和拓扑，第三部分

• 批准号: 1503701
• 负责人: Jeffrey Weeks
• 金额: $ 17.88万
• 依托单位: Weeks Jeffrey R
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-09-01 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1503701&HistoricalAwards=false
• 关键词: Geometry; Topology; Mainstream; Culture; Part

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.

几何游戏软件（将在这个项目中开发）采用了现代几何和拓扑的一些最基本的概念——即多连接空间、对称性和高维——并将它们带给广大的学生、艺术家和其他外行。数学教学传统上是非常苛刻的：数学家经常要求他们的学生按照数学家自己的条件来满足他们。目前的项目改变了这种安排，以他们自己的方式，在他们熟悉的环境中，如游戏或艺术作品中，为公众带来美丽的数学理念。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物体的幻觉，人们可以把它放在手掌上玩。在玩游戏或画画的乐趣中，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像欣赏文学和音乐一样欣赏现代几何；与此同时，希望这种“高级”几何思想的接触可能有助于准备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使其栩栩如生的软件而言，当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：Torus Games（八个熟悉的游戏，但在平面2流形和3流形上播放），KaleidoPaint（在17个欧几里得墙纸组的作用下创建彩色绘画），KaleidoTile（探索球面，平面和双曲表面上的多面体和瓷砖），Curved Spaces（球面，平面和双曲3流形的飞行模拟器）和4D Draw（4维矢量图形）。第二个目标是将Hyperbolic Games（基于双曲2流形的熟悉游戏）从一款小应用提升到旗舰级应用。第三个目标是开发两个全新的应用程序：4D Maze（4维空间的迷宫，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，考虑到人们普遍使用智能手机和平板电脑，为Android和iOS提供所有这些应用的版本是一个特别重要的任务。麦金塔和Windows的版本当然也会开发出来。