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

 Figure 8.10. Polyhedra with prismatic symmetry.

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

[1] P. R. Cromwell, Polyhedra, Cambridge University Press 1997.
[2] Martin Kraus' Live3D applet