Amplituhedra and origami, I: tree level

We establish a precise correspondence between points of the $m=4$ tree momentum amplituhedron and origami crease patterns. As an application, we prove that the BCFW (Britto-Cachazo-Feng-Witten) cells triangulate the $m=4$ tree amplituhedron both in momentum space and in momentum-twistor space. As another application, we show that every nondegenerate weighted planar bipartite graph $\mathrm{\Gamma }$ admits a t-embedding, i.e., an embedding of the planar dual of $\mathrm{\Gamma }$ such that the sum of angles of white (equivalently, black) faces around each vertex is equal to $\pi$.