# History

### of crystallographic groups and related topics

People have always been interested in patterns, both planar patterns and spacial patterns. 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. 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.

Kepler found other nearly regular solids and noted the regular tessellations (tilings) of the plane. There are only three regular tessellations, one of triangles, one of squares, and one of hexagons. There are also several nearly regular tessellations analogous to the Archimedean solids.

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.

#### Classification in two and three dimensions

In the nineteenth century the classification of planar and spacial lattices and patterns began. One of the problems was, of course, deciding when different patterns exhibited the same sort of regularity. A variety of classification methods were developed. At first, lattice structures were studied. Later, symmetries, and the way the symmetries were related, were used to make finer distinctions. The lattices were generally analyzed by means of quadratic forms using two variables in the planar case and three variables in the spacial case.
 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. J. F. C. Hessel. Kristall. Gehler's Physikalische Wöterbuch, pp. 1023-1360. Schwikert, Leipzig, 1830. Reprinted in Ostwald's Klassiker der Exakten Wissenschaften. Engelmann, Leipzig, 1897. All of the symmetrical networks of points which can have crystallographic symmetry were found geometrically by Frankenheim in 1835. Frankenheim. Die Lehre von der Cohösion. Breslau, 1835. O. Rodriques. 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. Jour. de Math. 5 (1840), 380-440. This theory of crystal classes was systematized by A. Bravais who classified the 14 types of spacial lattices. See also A. Gadolin (1871), and P. Curie (1884). A. Schoenflies described them with group theory in 1891 (see below). A. Bravais. Mé&moire sur les systèmes formé par des points distribués régulièrement sur un plan ou dans l'espace. J. Ecole Polytech. 19 (1850), 1-128. English translation: Memoir 1, Crystallographic Society of America, 1949. Also Monograph 4, American Crystallographic Association. A. Bravais. J. Ecole Polytech. 20 (1851), 102. Dirichlet described what has since been known as Dirichlet regions for lattices. (Need information on Voronoi.) G. Lejeune Dirichlet. Über die Reduction der positiven quadratischen Formen mit drei unbestimmten ganzen Zahlen. Journal für die reine und angewandte Mathematik 40 (1850), 209-227. G. Eisenstein. Tabelle der deducirten positiven ternären quadritischen Formen, nebst den Resultaten neuer Forschungen über diese Formen, in besonderer Rücksicht auf ihre tabellarische Berechnung. J. Reine Angew. Math. 41 (1851), 141-190. A. F. Möbius. Über das Gesetz der Symmetrie der Kristalle. J. Reine Angew. Math. 43 (1852), 365-?. Über symmetrische Figuren. Ibid. 44 (1852), 355-?. Also in Möbius's collected works II, pp. 349-372. 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. He listed 174 types of groups of motions including both crystallographic and nondiscrete groups. He also discovered 16 of the 17 wallpaper groups. Camille Jordan. Sur les groupes de mouvements. C. R. Acad. Sci. Paris 65 (1867), 229-232. Also Oeuvres de C. Jordan, vol. 4, pp. 113-116. Paris, Gauthier-Villars, 1964. C. Jordan. Mémoire sur les groupes de mouvements. Ann. Mat. Pur. App. (2) 2 (1868/1869), 167-215 and 322-345. Also Oeuvres, vol. 4, pages 231-302. C. Jordan. Mémoire sur l'équivalence des formes. J. Ecole Polytech. 48 (1880), 112-150. Also Oeuvres, vol. 3, pages 421-460. Starting in 1873 Sohncke applied Jordan's theory to two- and three-dimensional space, but the classification was incomplete. In 1879 he described the 65 types of rotational groups in space. Leonhard Sohncke. Die regelmässig ebenen Punkt systeme von unbegrenzter Ausdehnung.Borchardt J. 77, 47-102, and Berl. Monatsber. (1873), 578-583. Entwickelung einer Theorie der Krystallstruktur. Teubner, Leipzig, 1879. Entwickelung einer Theorie der Krystallstruktur. Z. Kryst. Min. 14 (1888), 417-425. Die Struktur der optisch drehenden Krystalle. ibid. 19 (1891), 529-559. E. Selling. Des formes quadratique binaires et ternaires. J. Math. Pures Appl. (3) 3 (1877), 21-60 and 153-207. 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. Fedorov 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. They were both completing a classification that had been going on for some time, and being a classification based on the same principles, they had to reach the same results. E. S. Fedorov. The elements of the study of figures. [Russian] Zapiski Imperatorskogo S. Peterburgskogo Mineralogichesgo Obshchestva [Proc. S. Peterb. Mineral. Soc.] (2) 21 (1885), 1-289. Symmetry of finite figures. [Russian] (2) 28 (1891), 1-146. Symmetry in the plane. [Russian] (2) 28 (1891), 345-390. E. S. Fedorov. Symmetry of Crystals. Translated from the 1949 Russian Edition. American Crystallographic Association, New York, 1971. A. M. Schoenflies. Über Gruppen von Bewegungen. Math. Ann. 28 (1886), 319-342, and 29 (1887), 50-80. Über Gruppen von Transformationen des Raumes in sich. ibid. 34 (1889), 172-203. Kristallsysteme und Kristallstruktur. Teubner, Leipzig, 1891. W. Barlow. Über die geometrische Eigenschaften homogener starrer Strukturen. Z. Kryst. Min. 23 (1894), 1-63. Nachtrag zu den tabellen. ibid. 25 (1896), 86-91. K. Rohn. Einige Sätze über regelmässige Punktgruppen. Math. Annalen, 53 (1900), 440-449. H. Hilton. Mathematical crystallography and the theory of groups of movements. Clarendon, 1903. G. Frobenius. Gruppentheoretische Ableitung der 32 Kristallklassen. S.-B. Preuss. Akad. Wiss. (1911), 681-691. (Gesammelte Abhandlungen. Springer, 1968, vol. 3, pp. 519-529.) P. Niggli. Geometrische Kristallographie des Diskontinuums. Gebrüder Borntraeger, 1919. G. Pólya and P. Niggli. Zeitschrift für Kristallographie und Mineralogie 69 (1924), 278-298. A standard for the names of the various two- and three-dimensional crystalography groups. N. F. M. Henry and K. Lonsdale. International Tables for X-Ray Crystallography, vol. 1, Kynoch Press, Birmingham, England, 1952.

#### Classification in higher dimensions

 The problem of understanding higher-dimensional crystallographic groups must have been fairly important, because it formed one part of problem 18 of the Hilbert problems. David Hilbert, a leading mathematician at the time, addressed the International Congress of Mathematicians in Paris in 1900. He proposed 23 fairly general problems to focus mathematical research. His question was "Is there in n-dimensional Euclidean space only a finite number of essentially different kinds of groups of motions with a fundamental region? The two- and three-dimensional cases were known, but no higher dimensional case. David Hilbert. Gött. Nachr. (1900), 253-297. English translation: Mathematical Problems. Bull. Amer. Math. Soc. 8 (1902), 437-479. On-line html version of the English translation. Bieberbach solved this problem in 1910. He proved that in any dimension that there were only finitely many groups. He didn't determine the actual number in any dimension, just that there were finitely many. Bieberbach showed that these groups were extensions of the translation group, which is isomorphic to Zn, by a finite subgroup of GL(n, Z). Now, cohomology groups are used to classify extensions. Also the groups we're considering are frequently called Bieberbach groups. L. Bieberbach. Über die Bewegungsgruppen des n-dimensionalen euklidischen Raumes mit einem endlichen Fundamentalbereich. Gött. Nachr.1910, 75-84. L. Bieberbach. Über die Bewegungsgruppen der Euklidischen Räume, I, Math. Ann. 70 (1911), 297-336, and II, Math. Ann. 72 (1912), 400-412. In 1948 Zassenhaus gave an algorithm to determine a complete set of representatives of the types of n-dimensional space groups. Über einen Algorithmus zur Bestimmung der Raumgruppen. Comment. Math. Helv. 21 (1948), 117-141. In 1974 there was a symposium at Northern Illinois University about the Hilbert problems, and John Milnor wrote about problem 18. J. Milnor. Hilbert's problem 18: on crystallographic groups, fundamental domains, and on sphere packing. Pages 491-506 in Mathematical Developments Arising from Hilbert Problems, edited by Felix Browder. American Mathematical Society, 1976. In the mid 1970's computers helped to determine that there are 4783 four-dimensional groups. Harold Brown, Rolf Bülow, Joachim Neubüser, Hans Wondratschek, and Hans Zassenhaus. Crystallographic Groups of Four-Dimensional Space. Wiley, New York, 1978. The actual number of groups for dimensions greater than four has not been determined, but there have been partial investigations. R. L. E. Schwarzenberger. N-dimensional Crystallography. Pitman, London, 1980.

#### Art

A number of art works from many cultures and ages depict many of the wallpaper patterns. One of the more recent artists that benefitted from the mathematical classification of them was M. C. Escher. From his friendship with H. S. M. Coxeter, Escher learned of various tessellations, particularly hyperbolic tessellations.