Discrete symmetries control geometric mechanics in parallelogram-based origami

James McInerney, Glaucio H. Paulino, D. Zeb Rocklin

Research output: Contribution to journalArticlepeer-review

2 Scopus citations


Geometric compatibility constraints dictate the mechanical response of soft systems that can be utilized for the design of mechanical metamaterials such as the negative Poisson's ratioMiura-ori origami crease pattern.Here, we develop a formalism for linear compatibility that enables explicit investigation of the interplay between geometric symmetries and functionality in origami crease patterns. We apply this formalism to a particular class of periodic crease patterns with unit cells composed of four arbitrary parallelogram faces and establish that their mechanical response is characterized by an anticommuting symmetry. In particular, we show that the modes are eigenstates of this symmetry operator and that these modes are simultaneously diagonalizable with the symmetric strain operator and the antisymmetric curvature operator. This feature reveals that the anticommuting symmetry defines an equivalence class of crease pattern geometries that possess equal and opposite in-plane and out-of-plane Poisson's ratios. Finally, we show that such Poisson's ratios generically change sign as the crease pattern rigidly folds between degenerate ground states and we determine subfamilies that possess strictly negative in-plane or out-of-plane Poisson's ratios throughout all configurations.

Original languageEnglish (US)
Article numbere2202777119
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number32
StatePublished - Aug 9 2022

All Science Journal Classification (ASJC) codes

  • General


  • metamaterials
  • origami
  • symmetry


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