# 3D illustrations to Chapter 8 of Cromwell's Polyhedra

#### Izidor Hafner Tomislav Zitko Faculty of Electrical Engineering, University of Ljubljana Trzaska 25 , 1000 Ljubljana , Slovenia e-mail: izidor.hafner@fe.uni-lj.si

One way to improve teaching of stereometry is to give 3D illustrations to well known textbooks. As an example let us take Chapter 8 of Cromwell's Polyhedra. Chapter 8 is dealing with symmetry.

Systems of rotational symmetry Cyclic symmetry Figure 8.1. A rotation axis in a cyclic system

Dihedral symmetry  Figure 8.2. Principal and secondary axes in a dihedral system  Figure 8.3. When n is even, the secondary axes in Dn can be separated into two kinds  Figure 8.4. Polyhedra with D2 symmetry.

Tetrahedral symmetry  Figure 8.5. Rotation axes in the tetrahedral system.

Octahedral symmetry   Figure 8.6. Rotation axes in the octahedral system.

Icosahedral Symmetry   Figure 8.7. Rotation axes in the icosahedral system.

Reflection symmetry Figure 8.9. A polyhedron with bilateral symmetry.

Prismatic symmetry types

Symmetry type Dnh. Figure 8.11.

Symmetry type Dnv. Figure 8.12.

Symmetry type Dn. Figure 8.13.

Symmetry type Cnv Figure 8.14.

Symmetry type Cnh Figure 8.15.

Symmetry type Cn. Figure 8.16.

Compound symmetry and the S2n symmetry type Figure 8.17.  reflection in a plane reflection in a point Figure 8.20.

Cubic symmetry types
Symmetry type Oh.  Figure 8.21. The reflection planes of a cube.

Symmetry type O Figure 8.22.

Symmetry type Th. Figure 8.23. Figure 8.25.

Symmetry type T Figure 8.26. Figure 2.27.

Some examples

The cube has octahedral rotational symmetry.

The dodecahedron has icosahedral rotational symmetry.

We get examples of tetrahedral symmetry by colouring polyhedra with octahedral and icosahedral symmetry.

References

 P. R. Cromwell, Polyhedra, Cambridge University Press 1997.
 Martin Kraus' Live3D applet