The symmetry of the seven crystal systems

Definition : A geometric figure is said to be symmetrical if it can be superimposed on itself by a symmetry operation. There are three symmetry operators.

Centre of symmetry (C) : A geometric figure has a centre of symmetry if each face is reproduced by an identical and parallel face on the opposite side. The centre corresponds to the intersection of the axes and planes of symmetry.

Plane of symmetry (or mirror) (M) : A plane of symmetry is a mirror which makes any point A of the geometric figure correspond to another point A’ located on the normal to this plane passing through A. A and A’ are equidistant from the plane.

Axis of symmetry (A) : A straight line is an axis of symmetry of order n when it is possible to rotate the geometrical figure around this line by an angle equivalent to 360°/n and the figure becomes confused with itself. In mineralogy, we find axes of order 2, 3, 4 and 6.

This article is complementary to the article on the geometry of crystal systems.

Symmetry of the cubic system

Reminders of the characteristics of the crystalline mesh

a=b b=c a=c
α=90° β=90° γ=90°

Symmetry of the quadratic system

Reminders of the characteristics of the crystalline mesh

a=b b≠c a≠c
α=90° β=90° γ=90°

Symmetry of the orthorhombic system

Reminders of the characteristics of the crystalline mesh

a≠b b≠c a≠c
α=90° β=90° γ=90°

Symmetry of the monoclinic system

Reminders of the characteristics of monoclinic mesh

a≠b b≠c a≠c
α=90° β≠90° γ=90°

Symmetry of the triclinic system

Reminders of the characteristics of triclinic mesh

a≠b b≠c a≠c
α≠90° β≠90° γ≠90°
α≠β

Symmetry of the rhombohedral system

Reminders of the characteristics of the rhombohedral mesh

a=b b=c a=c
α≠90° β≠90° γ≠90°
α=β=γ

Symmetry of the hexagonal system

Reminders of the characteristics of the mesh

a=b b≠c a≠c
α=90° β=90° γ=120°

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