The revival of the Bravais lattice

The following short quiz will test your knowledge and understanding of the Bravais lattice.

(1) What are the Bravais-lattice symbols of the following space groups: No. 2, No. 40, No. 150 and No. 161?

(2) What is the essential geometric difference between the Bravais-lattice types mP and mS? Note: The correct answer contains neither the words 'affine equivalent' nor the words 'unit cell'. Try also the Bravais-lattice types oP, oS, oF and oI.

The 14 Bravais-lattice types are at the very heart of crystallography. It is somewhat remarkable that, in the second decade of the 21st Century, we may still learn new things about them. In Grimmer's paper [Grimmer, H. (2015). Acta Cryst. A71, doi:10.1107/S2053273314027351] he does just this and provides important new insights. Grimmer presents an entirely original way of determining the hierarchical arrangement of Bravais-lattice types. The result is summarised in an easily understood figure. In the figure, the Bravais-lattice type at the upper end of a line is a special case of the type at its lower end. Grimmer's approach to determining the hierarchy is to examine the group-subgroup relations amongst the space groups of the Bravais-lattice types. The latter are those (14) symmorphic space groups with the point group of a holohedry.

[Flack, H.D. (2015). Acta Cryst. A71, doi:10.1107/S2053273315002557]