Hidden symmetries generate rigid folding mechanisms in periodic origami

James McNerney, Bryan Gin-ge Chen, Louis Simon Theran, Christian Santangelo, Zeb Rocklin*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Citations (Scopus)
8 Downloads (Pure)


We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell–Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami’s vertices. This supports the recent result by Tachi [T. Tachi, Origami 6, 97–108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet.
Original languageEnglish
Pages (from-to)30252-30259
JournalProceedings of the National Academy of Sciences of the United States of America
Issue number48
Early online date16 Nov 2020
Publication statusPublished - 1 Dec 2020


  • Origami
  • Mechanisms
  • Rigid folding
  • Topological polarization


Dive into the research topics of 'Hidden symmetries generate rigid folding mechanisms in periodic origami'. Together they form a unique fingerprint.

Cite this