text: Reciprocal space (also called -space) provides a way to visualize the results of the Fourier transform of a spatial function. It is similar in role to the frequency domain arising from the Fourier transform of a time dependent function; reciprocal space is a space over which the Fourier transform of a spatial function is represented at spatial frequencies or wavevectors of plane waves of the Fourier transform. The domain of the spatial function itself is often referred to as real space. In physical applications, such as crystallography, both real and reciprocal space will often each be two or three dimensional. Whereas the number of spatial dimensions of these two associated spaces will be the same, the spaces will differ in their quantity dimension, so that when the real space has the dimension length (L), its reciprocal space will of inverse length, so L (the reciprocal of length).
  Reciprocal space comes into play regarding waves, both classical and quantum mechanical. Because a sinusoidal plane wave with unit amplitude can be written as an oscillatory term , with initial phase , angular wavenumber  and angular frequency , it can be regarded as a function of both  and  (and the time-varying part as a function of both  and ). This complementary role of  and  leads to their visualization within complementary spaces (the real space and the reciprocal space). The spatial periodicity of this wave is defined by its wavelength , where ; hence the corresponding wavenumber in reciprocal space will be .
  In three dimensions, the corresponding plane wave term becomes , which simplifies to  at a fixed time , where  is the position vector of a point in real space and now  is the wavevector in the three dimensional reciprocal space. (The magnitude of a wavevector is called wavenumber.) The constant  is the phase of the wavefront (a plane of a constant phase) through the origin  at time , and  is a unit vector perpendicular to this wavefront. The wavefronts with phases , where  represents any integer, comprise a set of parallel planes, equally spaced by the wavelength .