Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-08T02:26:33.720Z Has data issue: false hasContentIssue false
  • English
  • Français

Crystallography in spaces of arbitrary dimension

Published online by Cambridge University Press:  24 October 2008

R. L. E. Schwarzenberger
R. L. E. Schwarzenberger
Affiliation:
University of Warwick
Get access Rights & Permissions [Opens in a new window]

Extract

The detailed study of n-dimensional crystallography appears to have been initiated by Hermann but unfortunately only a small fragment of his work was ever published(4). Recently, interest in the subject has been revived by the work of Bülow, Neubüser and Wondratschek(2), who not only obtained definitive results on 4-dimensional lattices and space groups but also proposed general definitions and procedures for the n-dimensional case. In particular they introduced the concept of crystal family and so resolved satisfactorily the controversy on whether there are six or seven 3-dimensional crystal systems (see(1)). Their point of view is arithmetic, that is, they use a coordinate system adapted to the particular lattice under consideration and represent elements of its symmetry group by integer matrices.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 76 , Issue 1 , July 1974 , pp. 23 - 32
Copyright
Copyright © Cambridge Philosophical Society 1974

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Buerger, M. J.Elementary crystallography, an introduction to the fundamental geometrical features of crytals (Wiley; New York, 1956).Google Scholar
(2)Bülow, R., Neubüser, J. and Wondratschek, H.On crystallography in higher dimension I, II, III. Acta Cryst. 27A (1971), 517–535.CrossRefGoogle Scholar
(3)Burckhardt, J. J.Die Bewegungsgruppen der Kristallographie (Birkhäuser Verlag; Basel, 1966).CrossRefGoogle Scholar
(4)Hermann, C.Kristallographie in Räumen beliebiger Dimensionzahl I. Acta Cryst. 2 (1949), 139–145; Translationsgruppen in n Dimensionzahl p. 24–30 of Zur, Struktur und Materie der Featkörper (Springer Verlag; Berlin–Göttingen–Heidelberg, 1951).CrossRefGoogle Scholar
(5)Kutosh, A. G.The theory of groups, vol. II (Chelsea; New York, 1960).Google Scholar
(6)Mozrzymas, J.Crystal symmetry groups and their representations I, II (preprint, Institut des Hautes Etudes Scientiflques, 1972).Google Scholar
(7)Phillips, F. C.An introduction to crystallography (Longmans; London, 1946).Google Scholar
(8)Schwarzenberger, R. L. E.Classification of crystal lattices. Proc.Cambridge Philos. Soc. 72 (1972), 325–349; The 17 plane symmetrygroups. Math. Gazette (to appear).CrossRefGoogle Scholar