Four-dimensional space groups for pedestrians: composite structures

Higher-dimensional crystals have been studied for the last thirty years. However, most practicing chemists, materials scientists, and crystallographers continue to eschew the use of higher-dimensional crystallography in their work. Yet it has become increasingly clear in recent years that the number of higher-dimensional systems continues to grow from hundreds to as many as a thousand different compounds. Part of the problem has to do with the somewhat opaque language that has developed over the past decades to describe higher-dimensional systems. This language, while well-suited to the specialist, is too sophisticated for the neophyte wishing to enter the field, and as such can be an impediment. This Focus Review hopes to address this issue. The goal of this article is to show the regular chemist or materials scientist that knowledge of regular 3D crystallography is all that is really necessary to understand 4D crystal systems. To this end, we have couched higher-dimensional composite structures in the language of ordinary 3D crystals. In particular, we developed the principle of complementarity, which allows one to identify correctly 4D space groups solely from examination of the two 3D components that make up a typical 4D composite structure.