Invariant and smooth limit of discrete geometry folded from bistable origami leading to multistable metasurfaces

Ke Liu, Tomohiro Tachi, Glaucio H. Paulino

Research output: Contribution to journalArticlepeer-review

92 Scopus citations


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.

Original languageEnglish (US)
Article number4238
JournalNature communications
Issue number1
StatePublished - Dec 1 2019
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • General Chemistry
  • General Biochemistry, Genetics and Molecular Biology
  • General Physics and Astronomy


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