Boris Gruber’s contributions to mathematical crystallography

What is it about?

Boris Gruber, a Czech mathematician, made fundamental contributions to the study of crystal lattices. There are 14 Bravais types of lattices. The three cubic types (primitive cP, face-centred cF, body-centred cI) have only one free parameter, the edge length a of the cubic cell. The hexagonal hP, the rhombohedral hR, and the two tetragonal types tP and tI have two free parameters a and c. The four orthorhombic Types oP, oC, oF, oI have three free parameters a, b and c. The two monoclinic types mP and mI have four free parameters a, b, c and an angle B. The anorthic type aP has six free parameters a, b, c and the angles a, B and x. As the number of free parameters is largest for anorthic lattices, it is most important to classify such lattices according to their crystallographic properties. In his paper Classification of lattices: a new step [1] Gruber defined 127 “Genera”, which constitute a finer classification than Bravais types. Whereas each of the three cubic Bravais types corresponds to one genus, each of the types aP and mI split into 43 genera. Lattices of the same genus agree in the densest directions and planes and the symmetry of these planes. For all Bravais types except aP Gruber determined necessary and sufficient conditions for the free parameters satisfied by the lattices corresponding to a given Bravais type [2].

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Photo by Daniele Levis Pelusi on Unsplash

Photo by Daniele Levis Pelusi on Unsplash

Why is it important?

In September 2022 a satellite “MACSMIN 2022” of the 33rd European crystallographic meeting was held in Liverpool, devoted to Crystal Lattice Classifications. The results [1] were an early step in this direction. Gruber also made the essential steps towards determining the limiting cases of Bravais types [2]. Poor health did not allow him finishing this task, which was completed by Grimmer [3]. Some of his results are also shown in [4]. References [1] Gruber, B. (1997). Acta Cryst. A 53, 505-521. [2] International Tables for Crystallography (2002), Volume A, 5th ed., edited by Theo Hahn. Section 9.3 “Further properties of lattices” by B. Gruber. [3] Grimmer, H. (2015). Acta Cryst. A 71, 143-149. [4] International Tables for Crystallography (2016), Volume A, 6th ed., edited by M. I. Aroyo. Section 3.1.4 “Further properties of lattices” by B. Gruber and H. Grimmer.