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  • Print publication year: 2012
  • Online publication date: November 2012

10 - Plane groups and space groups


The presentation of mathematics in schools should be psychological and not systematic. The teacher should be a diplomat. He must take account of the psychic processes in the boy in order to grip his interest, and he will succeed only if he presents things in a form intuitively comprehensible. A more abstract presentation is only possible in the upper classes.

Felix Klein, quoted in MacHale (1993)

In the previous chapter, we derived the 32 point group symmetries that are compatible with the translational symmetry of the 14 Bravais lattices. In the present chapter we ask the next logical question: what happens when we place a molecule (or a motif) with a certain point group symmetry G on each lattice node of a certain Bravais lattice J? We will show that this leads to the development of the 230 3-D crystallographic space groups; in 2-D, there are 17 plane groups. Furthermore, when time-reversal symmetry is included, the total numbers of plane and space groups increase dramatically to 80 and 1651, respectively. We conclude this chapter with a discussion of the use of space groups based on the International Tables for Crystallography.

Combining translations with point group symmetry

To answer the question above fully, we need to take every point group that belongs to a given crystal system and combine it with the translational symmetries of each of the Bravais lattices belonging to the same crystal system.

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