Three- and Four-Dimensional Triangulations and Mathematical Visualization

拓扑学是对几何对象的研究，其中忽略了长度和角度，但关注连通性。三角剖分是将曲面细分为三角形。类似地，我们将三维空间细分为四面体，并将高维空间细分为类似的高维几何形状。三角剖分是描述拓扑对象的最有效的方法之一，特别是在计算机中使用。有许多方法可以对拓扑对象进行三角剖分，对于特定的目的，每种方法都可能更好或更差。然而，不同的三角剖分可以通过简单的局部移动序列相互关联。这个由NSF资助的项目的中心目标之一是更好地了解三角测量的有用属性在我们通过这些移动改变它们时是如何改变的。另一个目标是数学可视化，以帮助研究、教学和推广。这包括使用新技术(包括3D打印、虚拟和增强现实)找到可视化数学对象的有效方法。PI开发了一门本科课程，将3D设计技能与生产3D打印对象所需的数学知识相结合。他计划将这种教学方法推广到定量科学的其他学科。PI正计划与同事们一起编写一本参考资料书，帮助其他人使用3D打印创建和教授数学。面向更广泛社区的外联活动将包括说明性论文、公开演讲、YouTube视频、开源可视化应用程序以及与数学博物馆的合作。在这个由NSF资助的项目中，PI和他的合作者一起致力于研究三角剖分的类别，包括具有基本边或角度结构的三角剖分，1-有效的几何或转向三角剖分：这些类别与拓扑和几何不变量之间的关系，在这些类别中构建三角剖分的方法，以及与这些类别相对应的三角剖分的Pachner图的子图的结构。另一个目标是将三维三角剖分的性质和结果推广到四维三角剖分。使用的方法将在很大程度上是组合的，初级研究生和本科生都可以使用。一个可视化项目是找到拓扑对象的规范3D几何表示，以便可以3D打印模型。主题包括Seifert曲面、纽结补的纤丝和曲面的共形校正平铺。将使用代数描述和离散优化过程来生成几何图形。3D打印的其他项目包括研究和构建有趣的链接和其他机制。之前在实现3D双曲几何的虚拟现实模拟以及2D双曲几何与直线的乘积方面的工作已经成功地激励了数学家、物理学家和公众。PI计划将这项工作扩展到其他瑟斯顿几何图形和更远的地方，帮助其他研究人员在这些几何图形中可视化他们感兴趣的对象，并构建引人入胜的互动体验，使这些几何图形更容易向公众开放。最后，PI旨在实现交互式拓扑模拟，例如，允许用户物理操作虚拟球体，该虚拟球体的行为与球体外翻上下文中的行为相同。

Topology is the study of geometric objects, in which lengths and angles are ignored, but connectivity is paid attention to. A triangulation is a subdivision of a surface into triangles. Analogously, we subdivide a three-dimensional space into tetrahedra, and higher dimensional spaces into similar higher dimensional geometric shapes. Triangulations are one of the most effective ways to describe topological objects, particularly for use with computers. There are many ways to triangulate a topological object, each of which may be better or worse for a particular purpose. However, different triangulations can be related to each other by sequences of simple, local moves. One of the central goals of this NSF funded project is to better understand how useful properties of triangulations change as we alter them by these moves. Another goal centers on mathematical visualization to aid in research, pedagogy and outreach. This includes finding effective ways to visualize mathematical objects using new technologies, including 3D printing, virtual, and augmented reality. The PI has developed an undergraduate course integrating 3D design skills with the mathematics needed to produce 3D printed objects. He plans to extend this pedagogical method to other subjects in quantitative science. With colleagues, the PI is planning to write a resource book to help others create and teach mathematics with 3D printing. Outreach activities to the broader community will include expository papers, public talks, YouTube videos, open-source visualization apps, and collaboration with mathematics museums. In this NSF funded project, together with his collaborators, the PI aims to study classes of triangulations, including triangulations with essential edges or angle structures, 1-efficient, geometric or veering triangulations: relations between these classes and topological and geometric invariants, methods of constructing triangulations in these classes, and the structure of subgraphs of the Pachner graph of triangulations corresponding to these classes. Another aim is to generalize properties and results from three-dimensional to four-dimensional triangulations. The methods used will be largely combinatorial, and accessible to beginning graduate and undergraduate students. One visualization project is to find canonical 3D geometric representations of topological objects, so that models can be 3D printed. Subjects include Seifert surfaces, fibrations of knot complements, and conformally correct tilings of surfaces. Algebraic descriptions and discrete optimization processes will be used to generate geometry. Other projects in 3D printing include study and construction of interesting linkages and other mechanisms. Previous work in implementing virtual reality simulations of 3D hyperbolic geometry, and the product of 2D hyperbolic geometry with the line, has already been successful in inspiring mathematicians, physicists, and members of the public. The PI plans to extend this work to the other Thurston geometries and beyond, aid other researchers in visualizing objects they are interested in within these geometries, and construct engaging interactive experiences to make these geometries more accessible to the public. Finally, the PI aims to implement interactive topological simulations, for example to allow a user to physically manipulate a virtual sphere that behaves as in the context of sphere eversion.