ELEMENTS OF SYMMETRY
The faces of perfect crystals are arranged in a regular pattern. This regularity is termed the symmetry of the crystal. Any plane which divides a crystal into two halves so that these two halves are mirror images is called a plane of symmetry. A cube, for instance, has nine such planes. A crystal has a centre of symmetry if its every face has a corresponding face which is a mirror image on the opposite side. The third element of symmetry of a crystal is termed the axis of symmetry. This is an axis passing through the crystal around which it can be rotated so that it occupies the same position in space at least twice in one complete turn. An axis of symmetry may be two-, three-, four- or six-fold according to the number of times the crystal occupies its first position in a complete turn, and the angles of rotation which are required to place the crystal into this position are respectively 180, 120, 90 and 60 degrees. The axes of symmetry are expressions of rectangular, triangular, square or hexagonal cross-sections of the simple form of a crystal. We find, for instance, that a flat surface such as a floor can be completely covered with identical tiles only if these are rectangles (two-fold symmetry), cubes (four-fold symmetry), equilateral triangles (three-fold symmetry) or hexagons (six-fold symmetry).