Liu Ke, Tachi Tomohiro, Paulino Glaucio H
Department of Mechanical and Civil Engineering, California Institute of Technology, Pasadena, CA, 91125, USA.
Graduate School of Arts and Sciences, University of Tokyo, Tokyo, Japan.
Nat Commun. 2019 Sep 17;10(1):4238. doi: 10.1038/s41467-019-11935-x.
Origami offers an avenue to program three-dimensional shapes via scale-independent and non-destructive fabrication. While such programming has focused on the geometry of a tessellation in a single transient state, here we provide a complete description of folding smooth saddle shapes from concentrically pleated squares. When the offset between square creases of the pattern is uniform, it is known as the pleated hyperbolic paraboloid (hypar) origami. Despite its popularity, much remains unknown about the mechanism that produces such aesthetic shapes. We show that the mathematical limit of the elegant shape folded from concentrically pleated squares, with either uniform or non-uniform (e.g. functionally graded, random) offsets, is invariantly a hyperbolic paraboloid. Using our theoretical model, which connects geometry to mechanics, we prove that a folded hypar origami exhibits bistability between two symmetric configurations. Further, we tessellate the hypar origami and harness its bistability to encode multi-stable metasurfaces with programmable non-Euclidean geometries.
折纸提供了一种通过与比例无关且无损制造来编程三维形状的途径。虽然这种编程主要关注单个瞬态状态下的镶嵌几何形状，但在此我们提供了从同心褶皱正方形折叠出光滑鞍形的完整描述。当图案中方形折痕之间的偏移均匀时，它被称为褶皱双曲抛物面（hypar）折纸。尽管它很受欢迎，但关于产生这种美观形状的机制仍有许多未知之处。我们表明，从具有均匀或非均匀（例如功能梯度、随机）偏移的同心褶皱正方形折叠出的优雅形状的数学极限始终是双曲抛物面。使用我们将几何与力学联系起来的理论模型，我们证明了折叠的hypar折纸在两种对称构型之间表现出双稳态。此外，我们对hypar折纸进行镶嵌，并利用其双稳态来编码具有可编程非欧几里得几何形状的多稳态超表面。
