RUI: Configuration Spaces of Rigid Origami

RUI：刚性折纸的配置空间

• 批准号: 1906202
• 负责人: Thomas Hull
• 金额: $ 22万
• 依托单位: Western New England University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906202&HistoricalAwards=false
• 关键词: RUI; Configuration; Spaces; Rigid; Origami

Origami, the art of paper folding, has been practiced for centuries. The mathematics behind origami, however, is not yet fully understood. In particular, some origami models can be folded and unfolded in such a way that we could make the crease lines be hinges and the paper between them stiff like sheet metal. Such models are called rigidly flexible origami and have applications that span the physical and biological sciences, ranging from unfolding solar sails to collapsible heart stents. This project will add mathematical tools that allow industrial applications to employ cutting-edge research, from large-scale architectural structures to nano-scale robotics driven by origami mechanics. The tools from this project will help design self-foldable structures. Currently self-folding designs in engineering, architecture, and the biological sciences involve building physical models in a trial-and-error approach, wasting time and resources. The self-folding research provided by this project will allow designers to avoid pitfalls and tighten the design-to-realization process significantly. In addition to the research component, the PI shall organize a diverse range of educational activities including in-service teacher training and education, undergraduate mentoring and preparation for graduate school; high-school and undergraduate classes on the mathematics of folding; for the public, general-audience articles, lectures, and exhibitions. This will increase interest in STEM fields through the fun, hands-on nature of origami while simultaneously disseminating project results.The methods of this project involve a blend of practical experimentation with theory. Programmed self-foldability of structures will be achieved by trimming away undesired paths from the configuration space of all possible rigid foldings. One approach is to transform a given rigid folding of a crease pattern into a kinematically equivalent rigid folding with fewer degrees of freedom. The PI has proposed such a transform and will develop others. Key to all of this, however, is gaining a better understanding of rigid origami configuration spaces, which are algebraically complicated and not well understood. The project seeks to understand, and exploit, local-to-global behavior that is present in many known examples of rigid origami. In these examples approximating the configuration space near the origin (the unfolded state) leads to exact equations for the global configuration space. Formulating rigid origami configuration spaces in this way will add insight into the general field of flexible polyhedral surfaces, as well as provide the data needed to prove the feasibility of origami crease pattern transforms and design reliably self-foldable origami mechanisms.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.

折纸是折纸的艺术，已经存在了几个世纪。然而，折纸背后的数学原理还没有完全被理解。特别是，一些折纸模型可以折叠和展开，这样我们就可以使折痕线成为铰链，而它们之间的纸张像金属片一样僵硬。这种模型被称为刚性柔性折纸，其应用范围跨越物理和生物科学，从展开的太阳帆到可折叠的心脏支架。该项目将增加数学工具，允许工业应用程序使用尖端研究，从大规模建筑结构到由折纸机械驱动的纳米级机器人。这个项目的工具将有助于设计可自我折叠的结构。目前，工程、建筑和生物科学中的自折叠设计涉及以试错法建立物理模型，浪费时间和资源。该项目提供的自折叠研究将使设计师能够避免陷阱，并显著收紧从设计到实现的过程。除研究部分外，国际教育协会还将组织各种教育活动，包括在职教师培训和教育、本科生辅导和研究生院准备；高中和本科生折叠数学课程；面向公众的文章、讲座和展览。这将通过折纸的乐趣和动手性质增加对STEM领域的兴趣，同时传播项目结果。这个项目的方法包括实际实验和理论的结合。通过从所有可能的刚性折叠的构形空间中修剪掉不需要的路径，将实现结构的可编程自折叠。一种方法是将折痕图案的给定刚性折叠转换为具有较少自由度的运动学上等效的刚性折叠。PI已经提出了这样的转变，并将发展其他的转变。然而，所有这一切的关键是更好地理解刚性折纸配置空间，这在代数上是复杂的，而且没有被很好地理解。该项目试图了解并利用许多已知的僵化折纸例子中存在的局部到全球行为。在这些例子中，近似原点(展开状态)附近的配置空间将导致全局配置空间的精确方程。以这种方式形成刚性折纸配置空间将增加对灵活多面体表面的一般领域的洞察，并提供必要的数据来证明折纸折痕图案转换和设计可靠的自折叠折纸机制的可行性。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。

项目成果

Quasi-twisting convex polyhedra

拟扭曲凸多面体

• 发表时间: 2022
• 期刊: Proc. of the 34th Canadian Conference on Computational Geometry (CCCG 2022
• 作者: Hull, T.;Lubiw, A.;O'Rourke, J.;Mundilova, K.;Nara, C.;Tkadlec, J.;Uehara, R.
• 通讯作者: Uehara, R.

Rigid Foldability is NP-Hard

刚性可折叠性是 NP 难的

• 发表时间: 2020
• 期刊: Journal of Computational Geometry
• 影响因子: 0.3
• 作者: H. A. Akitaya;E. D. Demaine;T. Horiyama;T. C. Hull;J. S. Ku;T. Tachi,
• 通讯作者: T. Tachi,

Maximal origami flip graphs of flat-foldable vertices: properties and algorithms

可平折叠顶点的最大折纸翻转图：属性和算法

• DOI: 10.7155/jgaa.00605
• 发表时间: 2022
• 期刊: Journal of Graph Algorithms and Applications
• 作者: Hull, Thomas C.;Morales, Manuel;Nash, Sarah;Ter-Saakov, Natalya
• 通讯作者: Ter-Saakov, Natalya

Folding points to a point and lines to a line

将点折叠为点，将线折叠为线

• 发表时间: 2021
• 期刊: Proceedings of the 33rd Canadian Conference on Computational Geometry (CCCG 2021
• 作者: Akitaya, Hugo A.;Ballinger, Brad;Demaine, Erik D.;Hull, Thomas C.;Schmidt, Christiane
• 通讯作者: Schmidt, Christiane

Explicit kinematic equations for degree-4 rigid origami vertices, Euclidean and non-Euclidean

4 度刚性折纸顶点、欧几里德和非欧几里德的显式运动方程

• DOI: 10.1103/physreve.106.055001
• 发表时间: 2022
• 期刊: Physical Review E
• 影响因子: 2.4
• 作者: Foschi, Riccardo;Hull, Thomas C.;Ku, Jason S.
• 通讯作者: Ku, Jason S.