  A complete characterization is given of the possible macroscopic deformations
of periodic nonlinear affine unimode metamaterials constructed from rigid bars
and pivots. The materials are affine in the sense that their macroscopic
deformations can only be affine deformations: on a local level the deformation
may vary from cell to cell. Unimode means that macroscopically the material can
only deform along a one dimensional trajectory in the six dimensional space of
invariants describing the deformation (excluding translations and rotations).
We show by explicit construction that any continuous trajectory is realizable
to an arbitrarily high degree of approximation provided at all points along the
trajectory the geometry does not collapse to a lower dimensional one. In
particular, we present two and three dimensional dilational materials having an
arbitrarily large flexibility window. These are perfect auxetic materials for
which a dilation is the only easy mode of deformation. They are free to dilate
to arbitrarily large strain with zero bulk modulus.
