We establish a bijection between points in the momentum amplituhedron and origami crease patterns. As an application, we prove the conjecture that the cells appearing in the Britto–Cachazo–Feng–Witten (BCFW) recursion form a triangulation of the momentum amplituhedron. As another application, we show that any weighted planar bipartite graph can be geometrically realized as an origami crease pattern, thereby confirming the conjecture of Chelkak–Laslier–Russkikh, which originated from the works of Kenyon and Smirnov on the conformal invariance of the dimer and Ising models.