Geometry and Topology in Mainstream Culture, Part III

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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.