Tables for
Volume A1
Symmetry relations between space groups
Edited by Hans Wondratschek and Ulrich Müller

International Tables for Crystallography (2006). Vol. A1, ch. 1.1, p. 2   | 1 | 2 |

Section 1.1.1. The fundamental laws of crystallography

Mois I. Aroyo,a* Ulrich Müllerb and Hans Wondratschekc

aDepartamento de Física de la Materia Condensada, Facultad de Ciencias, Universidad del País Vasco, Apartado 644, E-48080 Bilbao, Spain,bFachbereich Chemie, Philipps-Universität, D-35032 Marburg, Germany, and cInstitut für Kristallographie, Universität, D-76128 Karlsruhe, Germany
Correspondence e-mail:

1.1.1. The fundamental laws of crystallography

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The documentation concerning group-theoretical aspects of group–subgroup relations and the rising importance of these relations in three-dimensional crystallography is scattered widely in the literature. This short review, therefore, cannot be exhaustive and may even be unbalanced if the authors have missed essential sources. Not included here is the progress made in the general theory of crystallographic groups, e.g. in higher dimensions, which is connected with the names of the mathematicians and physicists Ascher, Brown, Janner, Janssen, Neubüser, Plesken, Zassenhaus and others. This volume A1 of International Tables for Crystallography, abbreviated IT A1, is concerned with objects belonging to the `classical' theory of crystallographic groups.

For a long time, the objective of crystallography, as implied by the word itself, was the description of crystals which were found in nature or which grew from solutions of salts or from melts. Crystals often display more-or-less planar faces which are symmetrically equivalent or can be made so by parallel shifting of the faces along their normals such that they form a regular body. The existence of characteristic angles between crystal faces was observed by Niels Stensen in 1669. The importance of the shape of crystals and their regularity was supported by observations on the cleavage of crystals. It was in particular this regularity which attracted mineralogists, chemists, physicists and eventually mathematicians, and led to the establishment of the laws by which this regularity was governed. The law of symmetry and the law of rational indices were formulated by René Just Haüy around 1800. These studies were restricted to the shape of macroscopic crystals and their physical properties, because only these were accessible to measurements until the beginning of the twentieth century.

Later in the nineteenth century, interest turned to `regular systems' of points, for which the arrangement of points around every point is the same. These were studied by Wiener (1863[link]) and Sohncke (1879[link]). For such sets, there are isometric mappings of any point onto any other point which are such that they map the whole set onto itself. The primary aim was the classification and listing of such regular systems, while the groups of isometries behind these systems were of secondary importance. These regular systems were at first the sets of crystal faces, of face normals of crystal faces and of directions in crystals which are equivalent with respect to the physical properties or the symmetry of the external shape of the crystal. The listing and the classification of these finite sets was based on the law of rational indices and resulted in the derivation of the 32 crystal classes of point groups by Moritz Ludwig Frankenheim in 1826, Johann Friedrich Christian Hessel in 1830 and Axel V. Gadolin in 1867, cited by Burckhardt (1988[link]). The list of this classification contained crystal classes that had not been observed in any crystal and included group–subgroup relations implicitly.

Crystal cleavage led Haüy in 1784 to assume small parallel­epi­peds as building blocks of crystals. In 1824, Ludwig August Seeber explained certain physical properties of crystals by placing molecules at the vertices of the parallelepipeds. The concepts of unit cell and of translation symmetry were thus introduced implicitly.

The classification of the 14 underlying lattices of translations was completed by Auguste Bravais (1850[link]). In the second half of the nineteenth century, interest turned to the derivation and classification of infinite regular systems of points or figures, in particular by Leonhard Sohncke and Evgraf Stepanovich Fedorov. Sohncke, Fedorov (1891[link]), Arthur Schoenflies (1891[link]) and later William Barlow (1894[link]) turned to the investigation of the underlying groups, the space groups and plane groups. The derivation and classification of these groups were completed in the early 1890s. It was a plausible hypothesis that the structures of crystals were related to combinations of regular systems of atoms (Haag, 1887[link]) and that the symmetry of a crystal structure should be a space group, but both conjectures were speculations at that time with no experimental proof. This also applies to the atom packings described by Sohncke and Barlow, such as a model of the NaCl structure (which they did not assign to any substance).


Barlow, W. (1894). Über die geometrischen Eigenschaften homogener starrer Strukturen. Z. Kristallogr. Mineral. 23, 1–63.Google Scholar
Bravais, A. (1850). Mémoire sur les systèmes formés par les points distribués régulièrement sur un plan ou dans l'espace. J. Ecole Polytech. 19, 1–128. (English: Memoir 1, Crystallographic Society of America, 1949.)Google Scholar
Burckhardt, J. J. (1988). Die Symmetrie der Kristalle. Basel: Birkhäuser.Google Scholar
Fedorov, E. (1891). Symmetry of regular systems and figures. Zap. Mineral. Obshch. (2), 28, 1–46. (In Russian.) (English: Symmetry of crystals, American Crystallographic Association, 1971.)Google Scholar
Haag, F. (1887). Die regulären Krystallkörper. Programm des Königlichen Gymnasiums in Rottweil zum Schlusse des Schul­jahres 1886–1887.Google Scholar
Schoenflies, A. M. (1891). Krystallsysteme und Krystallstruktur. Leipzig: Teubner. Reprint 1984: Springer.Google Scholar
Sohncke, L. (1879). Entwickelung einer Theorie der Krystallstruktur. Leipzig: Teubner.Google Scholar
Wiener, C. (1863). Grundzüge der Weltordnung. I. Atomlehre, p. 82. Leipzig: Heidelberg.Google Scholar

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