The revival of the Bravais lattice.

The 14 Bravais-lattice types are at the very heart of crystallography, learned in Chapter 1 of any introductory course on the topic. It is somewhat remarkable that, in the second decade of the 21st century, we may still learn new things about them, but Hans Grimmer's paper Partial order among the 14 Bravais types of lattices: basics and applications (Grimmer, 2015) does this and provides us with important new insights. It presents an entirely original way of determining the hierarchical arrangement of Bravais-lattice types. The result is summarized in an easily understood figure (Fig. 2 in the paper, reproduced here also as Fig. 2). 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. Grimmer's approach yields the same result as previous analyses based on consideration of the metrical properties of lattices by way of special cases of the unit-cell dimensions. The hierarchy of the Bravais lattices is largely unknown and ignored by the crystallographic community. Implicitly, this hierarchy, and indeed the Bravais lattices themselves, are apparently considered to be of little importance and use to crystal-lographers. This is startling, as a fundamental property of a space group is its set of translational symmetry operations represented by the Bravais lattice. Grimmer's paper shows us how knowledge of the Bravais lattice hierarchy can help us solve real-world structures. 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 (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. which is reproduced here as Fig. 1. It shows the classifications of space groups. In the upper part of the diagram, the classification splits into two separate branches. On the right-hand branch, there is a classification in terms of the point group of the space group indicated as the '32 (geometric) crystal classes'. This classification is greatly loved by mineralogists and those interested in the physical properties of crystals. The banner …