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.
In the words of one of this proposal's anonymous reviewers: "Mathematical concepts and ideas have traditionally been expressed in words and symbols, using an abstruse vocabulary that has been built up painstakingly over hundreds of years. For a traditionally trained mathematician this works well. But it takes a long time and is hard work for even a highly intelligent layperson to pick up enough of the lingo to become comfortable with it, and this sets up a very high barrier for non-mathematicians who would like to understand some part of modern mathematics." The present project's purpose was to strip away the abstruse vocabulary and formalism, and present a few of the most beautiful ideas of modern geometry and topology in a concrete, visual, even tacile form that students, artists and other laypeople can appreciate and enjoy. To achieve this goal, the P.I. selected three of the most important ideas, namely 1. symmetry 2. multiconnected spaces 3. 4-dimensional space and wrote computer software to simulate these beautiful objects so smoothly and realistically that users feel as if they are holding and manipulating the novel objects in the palms of their hands. Moreover, the software presents these simulations in a context that people are familiar with, like games or artwork. The user can immediately start playing a game in a multiconnected space, or drawing a picture with built-in symmetry, and gradually build up a gut-level intuition and understanding of the mathematical concept that it embodies. During the 2012-2014 period funded by this grant, the following new software was designed, written, field-tested, released and supported: - The "Torus Games" were extended to include multiconnected 3-dimensional spaces as well as 2-dimensional ones. - A "curve editor" was added to the symmetrical drawing app KaleidoPaint, so users may freely modify their drawings at any time. - The symmetrical drawing app KaleidoPaint was re-written to make use of newer operating system capabilities and provide good performance even for artists with over a thousand complex drawings. - The Move & Turn polyhedron game was adapted for museum use and is now running live in the main gallery of Heureka, the Finnish Science Centre. - An app "4D Maze" was written to give a truly broad audience a glimmer of 4D space, by letting players solve 4D mazes. - An app "4D Draw" was written to meet the more advanced needs of undergraduates (and curious adults) wanting to learn to visualize 4D space. - Given that most people are now more likely to seek recreational math software from a smartphone or tablet than from a desktop computer, the P.I. has shifted his programming emphasis to Android and iOS, while still of course maintaining versions for Windows and Macintosh. The P.I. wrote Android and iOS versions of several of the apps during the 2012-2014 period funded by this grant, and plans to complete Android and iOS versions of the remaining apps as soon as possible. - All apps have been maintained and revised to work well on current as well as older hardware. - The P.I. has continued his outreach activities as a "contact person" for geometry and topology. The apps' ultimate purpose is two-fold: First, to attract students to careers in mathematics and other STEM fields. Indeed the P.I. occasionally meets graduate students and faculty who tell him that his earlier writings and software were what made them decide on a career in mathematics. The apps' second, and equally important, purpose is to disseminate beautiful ideas from higher mathematics out into our broader culture, for the use and enjoyment of all. All the apps are available free of charge from www.geometrygames.org .
