text: Consider two lattice points A and B separated by a translation vector r. Consider an angle α such that a rotation of angle α about any lattice point is a symmetry of the lattice. Rotating about point B by α maps point A to a new point A.  Similarly, rotating about point A by α maps B to a point B.  Since both rotations mentioned are symmetry operations, A and B must both be lattice points.  Due to periodicity of the crystal, the new vector r which connects them must be equal to an integer multiple of r':
  with  integer. The four translation vectors, three of length  and one, connecting A and B, of length , form a trapezium.  Therefore, the length of r' is also given by:
  Combining the two equations gives:
  where  is also an integer. Bearing in mind that  we have allowed integers . Solving for possible values of  reveals that the only values in the 0° to 180° range are 0°, 60°, 90°, 120°, and 180°.  In radians, the only allowed rotations consistent with lattice periodicity are given by 2π/n, where n = 1, 2, 3, 4, 6.  This corresponds to 1-, 2-, 3-, 4-, and 6-fold symmetry, respectively, and therefore excludes the possibility of 5-fold or greater than 6-fold symmetry.