Kirigami, the art of paper cutting, has become the subject of study in mechanical metamaterials in recent years. The basic building blocks of any kirigami structures are repetitive deployable patterns, the design of which has to date largely relied on inspirations from art, nature, and intuition, embedded in a choice of the underlying pattern symmetry. Here we complement these approaches by clarifying some of the connections between kirigami patterns and symmetry in terms of the wallpaper groups, the set of seventeen plane symmetry groups that fully characterize the space of periodic tilings of the plane. We start by showing that deployable kirigami patterns in any of the seventeen wallpaper groups can be constructed, and then design symmetry-preserving cut patterns that can be effectively applied for achieving arbitrary size change throughout deployment. We further prove that different symmetry changes can be achieved by controlling the shape and connectivity of the tiles. Finally, we connect these results using the underlying lattice structures of the kirigami patterns. All together, our work provides a systematic approach for creating deployable structures with any prescribed size and symmetry properties, thereby paving the way for the design of a wide range of metamaterials by harnessing kirigami.

中文翻译：

---

壁纸组kirigami

剪纸艺术Kirigami近年来已成为机械超材料的研究主题。任何kirigami结构的基本构建块都是可重复部署的图案，迄今为止，其设计很大程度上依赖于艺术，自然和直觉的灵感，这些灵感嵌入在潜在的图案对称性选择中。在这里，我们通过阐明墙纸组中的折纸图案和对称性之间的一些联系来补充这些方法，墙纸组是十七个平面对称性组的集合，充分描述了平面的周期性平铺的空间。首先，我们说明可以构建17个壁纸组中任何一个中的可展开的折纸图案，然后设计保留对称性的切割模式，可以有效地应用该模式以在整个部署过程中实现任意大小的更改。我们进一步证明，通过控制瓦片的形状和连接性，可以实现不同的对称性更改。最后，我们使用kirigami模式的基础晶格结构连接这些结果。总之，我们的工作为创建具有任何规定大小和对称特性的可展开结构提供了一种系统化的方法，从而为利用奇异折纸设计各种超材料铺平了道路。