RUI: Configuration Spaces of Rigid Origami

折纸，折纸艺术，已经实践了几个世纪。然而，折纸背后的数学还没有完全理解。特别是，一些折纸模型可以折叠和展开，我们可以使折痕线成为铰链，它们之间的纸像金属片一样坚硬。这种模型被称为刚性柔性折纸，其应用范围涵盖物理和生物科学，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，使工业应用能够采用尖端研究，从大型建筑结构到由折纸力学驱动的纳米级机器人。该项目的工具将有助于设计自折叠结构。目前，工程、建筑和生物科学中的自折叠设计涉及以试错法构建物理模型，浪费时间和资源。该项目提供的自折叠研究将使设计人员能够避免陷阱，并显着收紧设计到实现的过程。除了研究部分，PI应组织各种各样的教育活动，包括在职教师培训和教育，本科生指导和研究生院的准备;高中和本科班的数学折叠;为公众，一般观众的文章，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，同时传播项目成果。该项目的方法涉及实践实验与理论的融合。通过从所有可能的刚性折叠的配置空间中修剪掉不需要的路径，将实现结构的程序化自折叠性。一种方法是将折痕图案的给定刚性折叠转换成具有较少自由度的运动学等效刚性折叠。PI已经提出了这样的转换，并将开发其他转换。然而，所有这些的关键是更好地理解刚性折纸构型空间，它在代数上很复杂，而且还没有得到很好的理解。该项目旨在理解和利用在许多已知的刚性折纸例子中存在的局部到全局行为。在这些例子中，近似原点附近的配置空间（展开状态）导致全局配置空间的精确方程。以这种方式制定刚性折纸配置空间将增加对柔性多面体表面的一般领域的洞察力，以及提供证明折纸折痕图案变换的可行性和设计可靠的自折叠折纸机制所需的数据。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.