seen as non-homogeneous METATOYs made from prisms; and frosted glass, which can be seen as an extreme case of an inhomogeneous, random METATOY made from prisms. Examples of METATOYs as defined above have existed long before analogies with metamaterials were noted and it was recognized that METATOYs can perform wave-optically forbidden ray-direction mappings (in pixellated form). Wave-optical constraints on light-ray fields and METATOYs Wave optics describes light at a more fundamental level than geometrical optics. In the ray-optics limit (in which the optical wavelength tends towards zero) of scalar optics (in which light is described as a scalar wave, an approximation that works well for paraxial light with uniform polarization), the light-ray field r corresponding to a light wave is its phase gradient, where is the phase of the wave . But according to vector calculus, the curl of any gradient is zero, that is and therefore This last equation is a condition, derived from wave optics, on light-ray fields. (Each of the three equations that makes up this vector equation expresses the symmetry of the second spatial derivatives, which is how the condition was initially formulated.) Using the example of ray-rotation sheets, it was shown that METATOYs can create light-ray fields that do not satisfy the above condition on light-ray fields. Relationship with metamaterials METATOYs are not metamaterials in the standard sense. The acronym "metamaterial for rays" was chosen because of a number of similarities between METATOYs and metamaterials, which are discussed below, along with the differences. In addition, metamaterials provided the inspiration for early METATOYs research, as summarized in the following quote: Motivated by the desire to build optical elements that have some of the visual properties of metamaterials on an everyday size scale and across the entire visible wavelength spectrum, we recently started to investigate sheets formed by miniaturized optical elements that change the direction of transmitted light rays. Similarities with metamaterials In a number of ways, METATOYs are analogous to metamaterials: structure: metamaterials are arrays of small (sub-wavelength size) wave-optical components (electro-magnetic circuits resonant with the optical frequency), whereas METATOYs are arrays of small (so that they work well as pixels), telescopic, "ray-optical components"; functionality: both metamaterials and METATOYs can behave like homogeneous materials, in the case of metamaterials a volume of material, in the case of METATOYs a sheet material, in both cases with very unusual optical properties such as negative refraction. Differences with metamaterials