  In this paper, we show that deciding rigid foldability of a given crease
pattern using all creases is weakly NP-hard by a reduction from Partition, and
that deciding rigid foldability with optional creases is strongly NP-hard by a
reduction from 1-in-3 SAT. Unlike flat foldability of origami or flexibility of
other kinematic linkages, whose complexity originates in the complexity of the
layer ordering and possible self-intersection of the material, rigid
foldability from a planar state is hard even though there is no potential
self-intersection. In fact, the complexity comes from the combinatorial
behavior of the different possible rigid folding configurations at each vertex.
The results underpin the fact that it is harder to fold from an unfolded sheet
of paper than to unfold a folded state back to a plane, frequently encountered
problem when realizing folding-based systems such as self-folding matter and
reconfigurable robots.
