The field of rigid origami concerns the folding of stiff, inelastic plates of material along crease lines that act like hinges and form a straight-line planar graph, called the crease pattern of the origami. Crease pattern vertices in the interior of the folded material and that are adjacent to four crease lines, i.e. degree-4 vertices, have a single degree of freedom and can be chained together to make flexible polyhedral surfaces. Degree-4 vertices that can fold to a completely flat state have folding kinematics that are very well-understood, and thus they have been used in many engineering and physics applications. However, degree-4 vertices that are not flat-foldable or not folded from flat paper so that the vertex forms either an elliptic or hyperbolic cone, have folding angles at the creases that follow more complicated kinematic equations. In this work we present a new duality between general degree-4 rigid origami vertices, where dual vertices come in elliptic-hyperbolic pairs that have essentially equivalent kinematics. This reveals a mathematical structure in the space of degree-4 rigid origami vertices that can be leveraged in applications, for example in the construction of flexible 3D structures that possess metamaterial properties.

翻译：刚性折纸领域研究的是沿折痕线折叠刚性、非弹性的材料板片，这些折痕线起到铰链作用，并形成直线平面图，称为折纸的折痕图案。位于折叠材料内部且与四条折痕线相邻的折痕图案顶点，即四度顶点，具有单一自由度，可以链接在一起形成柔性多面体表面。能够折叠到完全平坦状态的四度顶点，其折叠运动学已被充分理解，因此已在许多工程和物理应用中得到使用。然而，对于非平坦可折叠或并非由平坦纸张折叠而成的四度顶点，即顶点形成椭圆锥或双曲锥的情况，其折痕处的折叠角遵循更为复杂的运动学方程。在这项工作中，我们提出了一般四度刚性折纸顶点之间的一种新对偶关系，其中对偶顶点以椭圆-双曲对的形式出现，并具有本质上等效的运动学。这揭示了四度刚性折纸顶点空间中的一种数学结构，该结构可在应用中加以利用，例如在构建具有超材料特性的柔性三维结构中。