  Metamaterials are effectively homogeneous materials that display
extraordinary dispersion. Negative index metamaterials, zero index
metamaterials and extremely anisotropic metamaterials are just a few examples.
Instead of using locally resonating elements that may cause undesirable
absorption, there are huge efforts to seek alternative routes to obtain these
unusual properties. Here, we experimentally demonstrate an alternative approach
for constructing metamaterials with extreme dispersion by simply coiling up
space with curled channels. Such a geometric approach has the advantage that
the ratio between the wavelength and the lattice constant in achieving a
negative or zero index is easily tunable. It allows us to construct for the
first time an acoustic metamaterial with conical dispersion, leading to a clear
demonstration of negative refraction from an acoustic metamaterial with
airborne sound. We also design and realize a double-negative metamaterial for
microwaves under the same principle.
