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Three- and Four-Dimensional Triangulations and Mathematical Visualization

三维和四维三角测量和数学可视化

基本信息

批准号：
1708239
负责人：
Henry Segerman
金额：
$ 26.79万
依托单位：
Oklahoma State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-08-01 至 2021-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1708239&HistoricalAwards=false
关键词：
Three Four Dimensional Triangulations Mathematical

项目摘要

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.

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，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测量，每种方法对于特定目的都可能更好或更差。然而，不同的三角剖分可以通过简单的局部移动序列彼此关联。这个NSF资助的项目的中心目标之一是更好地理解三角测量的有用属性如何随着我们通过这些移动改变它们而改变。另一个目标是数学可视化，以帮助研究，教学和推广。这包括找到使用新技术（包括3D打印、虚拟和增强现实）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打印物体所需的数学相结合。他计划将这种教学方法推广到定量科学的其他学科。PI计划与同事一起编写一本资源书，以帮助其他人使用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临时论文、公开讲座、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NSF资助的项目中，PI与他的合作者一起研究三角剖分的类别，包括具有基本边缘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，以及对应于这些类别的三角剖分的Pachner图的子图的结构。另一个目的是推广性质和结果从三维到四维三角剖分。所使用的方法将主要是组合，并开始研究生和本科生访问。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。主题包括塞弗特表面，纤维化的结补充，并符合正确的瓷砖的表面。代数描述和离散优化过程将用于生成几何图形。3D打印的其他项目包括研究和构建有趣的联系和其他机制。以前在实现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Thurston几何形状及其他几何形状，帮助其他研究人员在这些几何形状中可视化他们感兴趣的对象，并构建引人入胜的交互式体验，使这些几何形状更容易为公众所接受。最后，PI旨在实现交互式拓扑模拟，例如允许用户物理操纵虚拟球体，其行为与球体外翻的情况相同。