Axes of inversion rotoinversion

Apart from the symmetry elements described in Chapter 3 and above, an additional type of rotation axis occurs in a solid that is not found in planar shapes, the inversion axis, n, (pronounced n bar ). The operation of an inversion axis consists of a rotation combined with a centre of symmetry. These axes are also called improper rotation axes, to distinguish them from the ordinary proper rotation axes described above. The symmetry operation of an improper rotation axis is that of rotoinversion. Two solid objects 

The operation of some of the other improper rotation axes can be illustrated with respect to the five Platonic solids, the regular tetrahedron, regular octahedron, regular icosahedron, regular cube and regular dodecahedron. These polyhedra have regular faces and vertices, and each has 

The tetrahedron illustrates the operation of a four-fold inversion axis, 4, (pronounced four bar ). A tetrahedron inscribed in a cube allows the three Cartesian axes to be defined 

The three-fold inversion axes relate the positions of the vertices to each other. A 3 inversion axis runs through the middle of each opposite 

Finally, it is seen that diads (2) run through Mirror  

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