The 14 types of Bravais lattices and their special cases

What is it about?

In three-dimensional space there are 14 Bravais types of lattices. All are special cases of the triclinic (anorthic) type. There are no types that are special cases of the three cubic types (primitive, body centred and all-face centred) or of the hexagonal type. So the question arises, which types are special cases of a given type?

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Why is it important?

If one determines a crystal structure, it is not always clear what lattice type it belongs to? Is it exactly hexagonal or has it a lower orthorhombic symmetry? This paper tells you for instance that the hexagonal lattice is a special case of only one of the four orthorhombic types and it gives necessary and sufficient metric conditions for the special cases in which a one-face centred orthorhombic lattice becomes hexagonal. The metric results described in the paper have been included as Section 3.1.4.4 in the 2016 edition of Volume A of the International Tables for Crystallography.