Boundary and Interface Modes in Periodically Triangulated Origami

Origami is an important system for architectured materials because its mechanical response is controlled by the geometry of its crease pattern.
While researchers are typically interested in the uniform deformations that arise from rigid folding, the role of boundaries and interfaces are important in design applications.
Here, we investigate the linear modes of triangulated origami, for which the number of constraints matches the number of degrees of freedom, i.e. mechanical criticality.
Such mechanically critical systems have zero modes on their boundaries which can be robustly moved from one side to the other, known as topological polarization.
However, Chen et al. (2016) found triangulated origami does not admit such topological polarization.
We first explain how this arises via a vertex duality in triangulated origami which prevents any net topological polarization.
We then explore triangulated origami as an element of a recently identified symmetry class which can admit new types of modes at interfaces that depend on the geometry of the joined crease patterns.