Geometric optimization and polygonal geometry

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This project involves the use of computer experimentation and graphical user interfaces to study simple but unsolved geometric problems. The PI is currently studying the paths taken by a superball as it bounces around the inside of a tetrahedral shaped room. The project will use computer experimentation to figure out how the patterns change as the room gradually changes shape. As another example, the PI is studying how bugs will crawl around the surface of a dodecahedron in order to move as far away from each other as possible. These simply defined problems exhibit a fascinating and sometimes unexpected complexity. These types of problems fall under the general rubric of geometric optimization and polygonal geometry. The project also involves training and mentoring of students, as well as dissemination via outreach and the creation of applets and computer programs to aid geometric visualization.In this project the PI will work on questions in geometry and dynamics which fall into two categories: geometric optimization and polygon geometry. The three main optimization topics that will be studied in this project are point configurations, paper Moebius bands, and the Lie group Sol. The three main polygon problems that will be investigated are polygonal outer billiards, symplectic polygonal billiards, and inscribed triangles and rectangles. Additionally, the PI plans to continue writing graphical user interfaces which help with research for the project, and the PI also plans to continue to write and illustrate children's math books.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.