Crystal systems are defined based on the symmetry of a crystal. There are six crystal systems, given in decreasing symmetry: cubic (or isometric), hexagonal, tetragonal, orthorhombic, monoclinic, and triclinic. Minimum symmetry requirements are: four 3-fold or -3 axes (cubic), one 3- or 6-fold axes (hexagonal), one 4-fold axis (tetragonal), three mutually perpendicular directions with 2-fold and/or mirror plane symmetries (orthorhombic), one 2-fold axis and/or mirror plane (monoclinic), and center of symmetry or identity operation only (triclinic). Consequently, because of the symmetry present, the relative lengths of the crystallographic axes and the values of interaxial angles may be constrained: cubic, a1 = a2 = a3, α = β = γ = 90o; hexagonal, a1 = a2 = a3 not equal c, α, β = 90o, γ = 120o, β = 90o; tetragonal, a1 = a2 not equal c, α = β = γ = 90o; orthorhombic, a not equal b not equal c, α = β = γ = 90o; monoclinic, a not equal b not equal c, α = γ = 90o, β > 90o; triclinic, a not equal b not equal c, α not equal β not equal γ. The term “isometric” is sometimes used as a morphological term where the measured crystal is equant.
Cf., crystallographic axes