  Traditional origami starts from flat surfaces, leading to crease patterns
consisting of Euclidean vertices. However, Euclidean vertices are limited in
their folding motions, are degenerate, and suffer from misfolding. Here we show
how non-Euclidean 4-vertices overcome these limitations by lifting this
degeneracy, and that when the elasticity of the hinges is taken into account,
non-Euclidean 4-vertices permit higher-order multistability. We harness these
advantages to design an origami inverter that does not suffer from misfolding
and to physically realize a tristable vertex.
