Amplituhedra and origami

We establish a precise correspondence between points of momentum amplituhedra and origami crease patterns. As an application, we prove that the BCFW cells triangulate the momentum amplituhedron when all Mandelstam variables are nonnegative. As another application, we show that every weighted planar bipartite graph $\Gamma$ admits a t-embedding, i.e., an embedding of the planar dual of $\Gamma$ such that the sum of angles of white (equivalently, black) faces around each vertex is equal to $\pi$.