  Deformations of conventional solids are described via elasticity, a classical
field theory whose form is constrained by translational and rotational
symmetries. However, flexible metamaterials often contain an additional
approximate symmetry due to the presence of a designer soft strain pathway.
Here we show that low energy deformations of designer dilational metamaterials
will be governed by a novel field theory, conformal elasticity, in which the
nonuniform, nonlinear deformations observed under generic loads correspond with
the well-studied conformal maps. We validate this approach using experiments
and finite element simulations and further show that such systems obey a
holographic bulk-boundary principle, which enables an unprecedented analytic
method to predict and control nonuniform, nonlinear deformations. This work
both presents a novel method of precise deformation control and demonstrates a
general principle in which mechanisms can generate special classes of soft
deformations.
