  Origami metamaterial design enables drastic qualitative changes in the
response properties of a thin sheet via the addition of a repeating pattern of
folds based around a rigid folding motion. Known also as a mechanism, this
folding motion will have a very small energy cost when applied uniformly; and
yet uniform activation of such remains highly difficult to observe, these
sheets instead generically displaying nonuniform response patterns which are
not yet well understood. Here, we present a purely geometric continuum theory
which captures the nonuniform, nonlinear response to generic loading as
composed locally of the planar mechanism, as well as previously identified
``twist'' and ``bend'' modes which enable the patterned sheet to curve out of
the plane across long distances. Our numerical analysis confirms that these
three modes govern the observed nonuniform response, varying smoothly across
the sheet according to three PDEs which guarantee compatibility. In analogy
with the recently solved case of planar mechanism metamaterials, these
``Orisometries'' (origami + isometries, so named by us) are subextensive but
infinite in number, with each mode displaying, in the linear limit, ``sheared
analytic'' spatial patterns which are controlled by the Poisson's ratio of the
uniform folding mechanism. Furthermore, the ``planar'' mechanism-based
deformation patterns superimpose with a mathematically dual space of
``non-planar'' twist/bend deformations to span the available soft linear
response. Together, our findings furnish the first quantification of the number
of soft response modes available, as well as the first intuitive quantification
of their spatial distribution.
