Folding a single high-genus surface into a repertoire of metamaterial functionalities.

The concepts of origami and kirigami have often been presented separately. Here, we put forth a synergistic approach-the folded kirigami-in which kirigami assemblies are complemented by means of folding, typical of origami patterns. Besides the emerging patterns themselves, the synergistic approach also leads to topological mechanical metamaterials.

Theorem on the Compatibility of Spherical Kirigami Tessellations.

We present a theorem on the compatibility upon deployment of kirigami tessellations restricted on a spherical surface with patterned slits forming freeform quadrilateral meshes. We show that the spherical kirigami tessellations have either one or two compatible states, i.e.