Hidden symmetries permit folding of all triangulated origami

Origami--thin sheets that fold along preset crease patterns--is at once an art form, an important technology for forming mechanical structures and a mathematically difficult problem of determining the folds permitted by a particular pattern. Origami devices are now being realized at the difficult-to-control atomic scale, motivating the question of which types of folding motions realized by crease patterns are also possible under broader classes of creases. We consider periodic triangulated origami of no engineered symmetry, and show that the geometry of the origami surface leads to hidden symmetries that link rigid-body motions of the sheet to force-bearing modes, which are then linked to folding motions. These folding patterns extend nonlinearly, permitting the origami to fold into cylindrical sections that can bend, twist and strain through particular configuration spaces. Our results also apply to related systems that, like triangulated origami, have balanced numbers of constraints and degrees of freedom, such as kirigami (cut origami) and continuum sheets, and can serve as the basis for a broad new class of deployable origami structures.