![]() In the late 19th century Fedorov, Schoenflies, and Barlow classified the 17 wallpaper groups (two-dimentional crystallography groups) and the 230 three-dimentional crystallography groups.įedorov wrote in his Symmetry of Crystals that, although he was familiar with Schoenflies' work, he claimed "a coincidence in the work of two researchers such as this has, perhaps, never before been observed in the history of science." This is a great exaggeration. Des formes quadratique binaires et ternaires. 14 (1888), 417-425.ĭie Struktur der optisch drehenden Krystalle. Teubner, Leipzig, 1879.Įntwickelung einer Theorie der Krystallstruktur. (1873), 578-583.Įntwickelung einer Theorie der Krystallstruktur. Die regelmässig ebenen Punkt systeme von unbegrenzter Ausdehnung. In 1879 he described the 65 types of rotational groups in space. Starting in 1873 Sohncke applied Jordan's theory to two- and three-dimensional space, but the classification was incomplete. He also discovered 16 of the 17 wallpaper groups.Ĭamille Jordan. He listed 174 types of groups of motions including both crystallographic and nondiscrete groups. In a general study of the theory of groups of movements Jordan gave described a general method for defining all of the possible ways of regularly repeating identical groupings of points. Über das Gesetz der Symmetrie der Kristalle. ![]() Tabelle der deducirten positiven ternären quadritischen Formen, nebst den Resultaten neuer Forschungen über diese Formen, in besonderer Rücksicht auf ihre tabellarische Berechnung. Journal für die reine und angewandte Mathematik 40 (1850), 209-227. Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. 20 (1851), 102.ĭirichlet described what has since been known as Dirichlet regions for lattices. Also Monograph 4, American Crystallographic Association.Ī. 19 (1850), 1-128.Įnglish translation: Memoir 1, Crystallographic Society of America, 1949. Mé&moire sur les systèmes formé par des points distribués régulièrement sur un plan ou dans l'espace. Schoenflies described them with group theory in 1891 (see below).Ī. Bravais who classified the 14 types of spacial lattices. This theory of crystal classes was systematized by A. Des lois géométriques qui régissent les déplacements d'un systeme solide dan l'espace, et de la variation des coordonées provenant de ces deplacements considérés indépendament des causes qui penvent les produire. Engelmann, Leipzig, 1897.Īll of the symmetrical networks of points which can have crystallographic symmetry were found geometrically by Frankenheim in 1835.įrankenheim. Reprinted in Ostwald's Klassiker der Exakten Wissenschaften. In 1831 Hessel first classified the 32 three-dimensional point groups (finite subgroups of the orthogonal group O(3) which correspond to the three-dimensional crystal classes. The lattices were generally analyzed by means of quadratic forms using two variables in the planar case and three variables in the spacial case. Later, symmetries, and the way the symmetries were related, were used to make finer distinctions. At first, lattice structures were studied. A variety of classification methods were developed. One of the problems was, of course, deciding when different patterns exhibited the same sort of regularity. In the nineteenth century the classification of planar and spacial lattices and patterns began. In the seventeenth century, Robert Hooke piled up "a company of bullets and some few other very simple bodies" to see the different ways that atoms could be arranged to build crystals, in particular, alum crystals.Ĭlassification in two and three dimensions There are also several nearly regular tessellations analogous to the Archimedean solids. There are only three regular tessellations, one of triangles, one of squares, and one of hexagons. Kepler found other nearly regular solids and noted the regular tessellations (tilings) of the plane. Archimedes generalized these to some nearly regular solids, now called Archimedean solids, such as the solid made out of pentagons and hexagons that is used for soccer balls and Buckyballs. Classification of patterns started two and a half millennia ago with the Pythagorean discovery that there are five regular solids: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. People have always been interested in patterns, both planar patterns and spacial patterns. Wallpaper Groups: history History of crystallographic groups and related topics
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