Isometric crystal system with symmetry elements

The isometric (cubic) system is the most symmetrical crystal system. It has three crystallographic axes of equal length, mutually perpendicular (all at 90°).

$$a = b = c, \qquad \alpha = \beta = \gamma = 90^\circ$$

        c
        |   b
        |  /
        | /
        |/____ a
   a = b = c, all at 90 deg

Symmetry elements (of the normal class, hexoctahedral)

• Planes of symmetry: 9 (3 axial planes + 6 diagonal planes).
• Axes of symmetry: 13
  • 3 tetrad (4-fold) axes through the centres of opposite faces,
  • 4 triad (3-fold) axes through opposite corners,
  • 6 diad (2-fold) axes through the mid-points of opposite edges.
• Centre of symmetry: 1.
• Total: 23 elements.

Common forms and minerals

• Cube, octahedron, dodecahedron.
• Minerals: galena, pyrite, fluorite, halite (rock salt), garnet, diamond, magnetite.