Geometric optimization and polygonal geometry

这个项目涉及使用计算机实验和图形用户界面来研究简单但未解决的几何问题。PI目前正在研究一个超级球在一个四面体形状的房间内反弹时所走的路径。该项目将使用计算机实验来计算出随着房间形状的逐渐改变，图案是如何变化的。作为另一个例子，PI正在研究虫子如何在十二面体的表面周围爬行，以便尽可能地远离彼此。这些简单定义的问题表现出一种迷人的、有时是意想不到的复杂性。这些类型的问题属于几何优化和多边形几何的一般范畴。该项目还包括对学生的培训和指导，以及通过外展传播和创建小程序和计算机程序来帮助几何可视化。在这个项目中，PI将研究几何和动力学问题，这两个问题分为两类：几何优化和多边形几何。本项目将研究的三个主要优化主题是点组态、纸Moebius带和Lie群Sol。将研究的三个主要多边形问题是多边形外部台球、辛多边形台球和内接三角形和矩形。此外，PI计划继续编写图形用户界面来帮助项目研究，PI还计划继续编写和说明儿童数学书籍。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.