  Inspired by the allure of additive fabrication, we pose the problem of
origami design from a new perspective: how can we grow a folded surface in
three dimensions from a seed so that it is guaranteed to be isometric to the
plane? We solve this problem in two steps: by first identifying the geometric
conditions for the compatible completion of two separate folds into a single
developable four-fold vertex, and then showing how this foundation allows us to
grow a geometrically compatible front at the boundary of a given folded seed.
This yields a complete marching, or additive, algorithm for the inverse design
of the complete space of developable quad origami patterns that can be folded
from flat sheets. We illustrate the flexibility of our approach by growing
ordered, disordered, straight and curved folded origami and fitting surfaces of
given curvature with folded approximants. Overall, our simple shift in
perspective from a global search to a local rule has the potential to transform
origami-based meta-structure design.

Soft Condensed Matter;Computational Geometry