# DoITPoMS

Demonstration of projection

While we have seen the basis of projection for a very simple system, we may now more closely examine the production of a stereogram (another word for stereographic projection), by producing the stereogram for a cube.

Look at a cube:

The most obvious symmetry element is the four-fold rotational symmetry. This presents on the sphere as a projected rotational axis, which intersects with the sphere, and is then projected down onto the projection plane.

It can easily be seen that the projected points of the rotational axes maintain the same symmetry and angular relationship on the projection as they do in 3D.

This can be extended further by the addition of 3-fold rotational axes which project from the vertices of the cube. (If these are not immediately obvious to you, please see the above image of a cube, and observe it down the [111] direction.)

To differentiate between the four-fold and three-fold rotational axes, we introduce a new notation, such that various types of symmetry elements, when projected onto the plane, are illustrated in different ways. These typically are:

So adding in the diads that project through edges of the cube with the new notation

The projection at this point is evidently getting rather complex. But the 2D representation is still easily understood. Here are all rotation axes on a single diagram.