Dissemination of IT for the Promotion of Materials Science (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.

There are also a collection of planes of symmetry. These intersect with the sphere in an infinite number of places, and as such, present as curves on the plane of projection, as can be seen here.

The curves here take the form of ‘great circles’. This will be covered in more detail later, but for now, all you need to know is that planes which pass through the origin (i.e. the centre of the sphere) present as great circles, which intersect with the primitive circle at the opposite ends of a diameter of the projection. In the full stereogram, you will also see that axes of rotation which lie in planes of symmetry show on the great circle.

So, assembling the entire stereogram, we see:

As you can see, the stereogram holds a large amount of information in a method which can be easily interpreted when you understand the principles behind it.