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A004031 Number of n-dimensional crystal systems.
(Formerly M3317)
1, 1, 4, 7, 33, 59, 251 (list; graph; refs; listen; history; text; internal format)



From Andrey Zabolotskiy, Jul 12 2017: (Start)

From Souvignier (2003): "the unions of all geometric classes intersecting the same set of Bravais flocks is defined to be a crystal system or point-group system. <...> This means that two geometric classes belong to the same crystal system if for any representative of the first class there is a representative of the other class such that the representatives have GL…(n,Q)†-conjugate Bravais groups. <...> The definition for crystal systems as given by Brown et al. (1978) therefore is only valid in dimensions up to 4, where it coincides with the more general definition adopted here."

For dimension 6, Souvignier (2003) uses old incorrect CARAT data, but the error affected only geometric classes and finer classification, so the data for crystal systems must be correct.

Among 33 4-dimensional crystal systems, 7 are enantiomorphic.

Coincides with the number of n-dimensional Bravais systems for n<5 (only).



P. Engel, "Geometric crystallography," in P. M. Gruber and J. M. Wills, editors, Handbook of Convex Geometry. North-Holland, Amsterdam, Vol. B, pp. 989-1041.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).


Table of n, a(n) for n=0..6.

H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978, p. 52. Corrections.

J. Neubüser, W. Plesken, and H. Wondratschek, An emendatory discussion on defining crystal systems, Commun. Math. Chem., 10 (1981), 77-96.

B. Souvignier, Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6, Acta Cryst., A59 (2003), 210-220.

W. Plesken and T. Schulz, CARAT Homepage

W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]


Cf. A004026-A004029, A004032, A006226, A006227, A080738.

Sequence in context: A241426 A271676 A149089 * A243863 A153062 A237424

Adjacent sequences:  A004028 A004029 A004030 * A004032 A004033 A004034




N. J. A. Sloane


a(5)-a(6) from Souvignier (2003) by Andrey Zabolotskiy, Jul 12 2017



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Last modified July 28 10:57 EDT 2017. Contains 289887 sequences.