Introduction to the symmetry of polyhedra

Regular polyhedra exhibit the property that they appear identical after some rotation. For example,  the cube can be rotated  by 90° or multiple of 90° around an axis passing through the middle of two parrallel square faces. In this transformation (also called a displacement), the cube before the rotation and after the rotation can not be distinguished. In other words the cube is invariant relative to a rotation of 90°. The set of all the transformations which leave a cube or any other object invariant are called the symmetry operations of an object. One particular aspect of these symmetry operations is that at least one point remains invariant.  Therefore the group is called the point symmetry group of the object.

In addition to rotation, they are other types of transformations which leave an object invariant, for example inversion, reflection or a combination of them. Fortunately they are only two types of transformation to consider, either a rotation or an improper rotation. We shall see that the inversion, the reflection or more generally the  roto-inversion are all improper rotations.

The purpose of this applet is to learn about proper and improper rotations. We shall simulate the rotation of objects directly on the screen by selecting first an object, second the axis of rotation an use a cursor to define the magnitude of the rotation. We shall see that the object will recover itself after a specific rotation. We shall also see that the objects presented here exhibit more than one rotation axis (this is not always the case). In order to generate an improper rotation, we shall combine two displacements,  a rotation and an inversion (or an inversion and a rotation). The inversion is the operation which transforms any point (xyz) to (-x-y-z) through the origin. The inversion is an operation which can be simulated on virtual object but not on real object without recreating a new object.

As an example of an improper rotation, let us consider the tetragonal bisphenoïd. A rotation of 90°about the -4 (4 bar) axis followed by an inversion will leave the object invariant. However, neither the inversion alone nor the 90° rotation alone will leave the object invariant. Such an operation is called a roto-inversion of 90° and is characterised by the symbol -4.    

The reader will wonder if we can generate a mirror operation based on rotation (proper or improper) only. This is indeed the case and can be nicely illustrated with the trigonal bipyramid. The combination of a rotation by 180° and the inversion yields precisely the reflection perpendicular to the rotation axis. The reader can convince himself by rotating the bipyramid along the three-fold axis by 180° followed by an inversion. In the particular case, the rotation of 180° is equivalent to a rotation of 60° owing to the threefold axis.

Practical details to perform the operations on the objects