NSF Award
2305250
Imagine a 3D surface made of many metal plates joined along their sides by hinges. Could such a surface start as a large dome and be flexed into a compact shape small enough to fit into a rocket? Could the flexing of such a surface be controlled to act as part of a robotics mechanism? Answering questions like these is the goal of this project. PI will develop new tools to establish a strong connection between general 3D flexible polyhedral surfaces (which might look like a dome, with positive curvature, or a saddle, with negative curvature) and origami, which is folded from flat, zero-curvature paper. The folding and unfolding of origami crease patterns has been studied heavily in recent years for applications in engineering and physics. Bringing mathematical tools from origami to flexible polyhedral surfaces could open up the field for practical applications in architecture, robotics, and structure designs for outer space. In addition, the PI will organize workshops and lectures on the topic of this project for students, educators, and the general public, leveraging the popularity of origami to increase interest in STEM and its intersections with art.<br/><br/>This project will investigate and develop three new tools to establish connections between flexible polyhedral surfaces and origami. The first is a newly-discovered dual relationship between vertices in a polyhedral surface that are elliptic (have positive discrete curvature) and hyperbolic (with negative curvature). PI will prove that such dual vertices of degree 4 are kinematically equivalent to each other (have the same kinematic equations) as well as to a family of degree-4 flat-foldable origami vertices, whose kinematics are very well-understood. The second is to establish a bijection between foldings of general origami vertices and flat-foldable origami vertices. The third is to find a geometric explanation for why it is so useful to parameterize the angles at each hinge of a flexible polyhedral surface with the tangent of the half angle; such parameterizations often linearize the configuration space of flexible polyhedral vertices, but little is known as to why. The techniques used to achieve these goals will include discrete differential geometry tools like the Gauss map and new tools from origami like the midpoint normal axes of a rigid origami vertex and projections of origami vertices into higher dimensional foldings.<br/><br/>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.
