text: In general, a geometric lattice is an infinite, regular array of vertices (points) in space, which can be modelled vectorially as a Bravais lattice. Some lattices may be skew, which means that their primary lines may not necessarily be at right angles. In reciprocal space, a reciprocal lattice is defined as the set of wavevectors  of plane waves in the Fourier series of any function  whose periodicity is compatible with that of an initial direct lattice in real space. Equivalently, a wavevector is a vertex of the reciprocal lattice if it corresponds to a plane wave in real space whose phase at any given time is the same (actually differs by  with an integer ) at every direct lattice vertex.
  One heuristic approach to constructing the reciprocal lattice in three dimensions is to write the position vector of a vertex of the direct lattice as , where the  are integers defining the vertex and the  are linearly independent primitive translation vectors (or shortly called primitive vectors) that are characteristic of the lattice. There is then a unique plane wave (up to a factor of negative one), whose wavefront through the origin  contains the direct lattice points at  and , and with its adjacent wavefront (whose phase differs by  or  from the former wavefront passing the origin) passing through . Its angular wavevector takes the form , where  is the unit vector perpendicular to these two adjacent wavefronts and the wavelength  must satisfy , means that  is equal to the distance between the two wavefronts. Hence by construction  and .
  Cycling through the indices in turn, the same method yields three wavevectors  with , where the Kronecker delta  equals one when  and is zero otherwise. The  comprise a set of three primitive wavevectors or three primitive translation vectors for the reciprocal lattice, each of whose vertices takes the form , where the  are integers. The reciprocal lattice is also a Bravais lattice as it is formed by integer combinations of the primitive vectors, that are , , and  in this case. Simple algebra then shows that, for any plane wave with a wavevector  on the reciprocal lattice, the total phase shift  between the origin and any point  on the direct lattice is a multiple of  (that can be possibly zero if the multiplier is zero), so the phase of the plane wave with  will essentially be equal for every direct lattice vertex, in conformity with the reciprocal lattice definition above. (Although any wavevector  on the reciprocal lattice does always take this form, this derivation is motivational, rather than rigorous, because it has omitted the proof that no other possibilities exist.)
  The Brillouin zone is a primitive cell (more specifically a Wigner–Seitz cell) of the reciprocal lattice, which plays an important role in solid state physics due to Bloch's theorem. In pure mathematics, the dual space of linear forms and the dual lattice provide more abstract generalizations of reciprocal space and the reciprocal lattice.