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
email: izidor.hafner@fe.unilj.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