A006227 -id:A006227 - cp's OEIS frontend

A307289 Erroneous version of A006227.

Original entry on oeis.org

2, 17, 230, 4895

Offset: 1

dead

Schwarzenberger, R. L. E. "Colour symmetry." Bulletin of the London Mathematical Society 16.3 (1984): 216-229.

A004029 Number of n-dimensional space groups.

Original entry on oeis.org

1, 2, 17, 219, 4783, 222018, 28927915

Offset: 0

N. J. A. Sloane

Right border of A293060. - Andrey Zabolotskiy, Oct 07 2017

H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978, p. 52.
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.
J. E. Goodman and J. O'Rourke, editors, Handbook of Discrete and Computational Geometry, CRC Press, 1997, p. 102 and 934.
T. Janssen, Crystallographic Groups. North-Holland, Amsterdam, 1973, p. 119.
R. L. E. Schwarzenberger, N-Dimensional Crystallography. Pitman, London, 1980, p. 34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Dror Bar-Natan, Illustrations of 2-dimensional symmetry groups
Carlos Cid and Tilman Schulz, Computation of Five and Six Dimensional Bieberbach Groups, Experimental Mathematics 10:1 (2001), 109-115.
W. Plesken and T. Schulz, CARAT Homepage
W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
W. Plesken and T. Schulz, Introduction to CARAT [Cached copy in pdf format (without subsidiary pages), with permission]
W. Plesken and T. Schulz, Counting crystallographic groups in low dimensions, Experimental Mathematics, 9 (No. 3, 2000), 407-411.
E. S. Rosenthal & N. J. A. Sloane, Correspondence, 1975
R. L. E. Schwarzenberger, Colour symmetry, Bulletin of the London Mathematical Society 16.3 (1984): 216-229.
N. A. Vavilov, Saint Petersburg School of the Theory of Linear Groups. I. Prehistory, Vestnik St. Petersburg Univ. (Russia 2023), Vol. 56, 273-288.
Wikipedia, Space group
Index entries for sequences related to groups

Cf. A006227, A004027, A293060.

a(6) corrected by W. Plesken and T. Schulz. Thanks to Max Horn for reporting this correction, Dec 18 2009

A004027 Number of arithmetic n-dimensional crystal classes.

Original entry on oeis.org

1, 2, 13, 73, 710, 6079, 85308

Offset: 0

N. J. A. Sloane

Number of Z-classes of finite subgroups of GL_n(Z) up to conjugacy.

H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978, p. 52.
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.
R. L. E. Schwarzenberger, N-Dimensional Crystallography. Pitman, London, 1980, p. 34.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Akinari Hoshi, Ming-chang Kang and Aiichi Yamasaki, Multiplicative Invariant Fields of Dimension <= 6, Memoirs of the AMS, 2023.
W. Plesken and T. Schulz, Dominik Bernhardt and others, Computer package CARAT
W. Plesken and T. Schulz, CARAT Homepage [dead link]
W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
W. Plesken and T. Schulz, Introduction to CARAT [Cached copy in pdf format (without subsidiary pages), with permission]
W. Plesken and T. Schulz, Counting crystallographic groups in low dimensions, Experimental Mathematics, 9 (No. 3, 2000), 407-411.
R. L. E. Schwarzenberger, Colour symmetry, Bulletin of the London Mathematical Society 16.3 (1984): 216-229.

Cf. A004028, A004029, A006227, A307288.

a(6) corrected from CARAT page by D. S. McNeil, Jan 02 2011

A293061 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, counting enantiomorphs.

Original entry on oeis.org

1, 2, 2, 10, 7, 17, 32, 75, 80, 230, 271, 343, 1091, 1594, 4894, 955

Offset: 0

Andrey Zabolotskiy, Sep 29 2017

T(n,0) count n-dimensional crystallographic point groups, T(n,n) count n-dimensional space groups (i.e., right border is A006227). The name "subperiodic groups" is usually related to the case 0 < k < n only, i.e., symmetry groups of n-dimensional objects including k independent translations which are subgroups of some n-dimensional space groups.
The Bohm symbols for these groups are G_{n,k}, except for the case k=n, when it is G_n.
Some groups have their own names:
T(2,1): frieze groups
T(2,2): wallpaper groups
T(3,1): rod groups
T(3,2): layer groups
[Palistrant, 2012, p. 476] gives correct T(4,k), k=0,1,2,3 but incorrect T(4,4). For correct value of T(4,4), see [Souvignier, 2006, p. 80].

			The triangle begins:
    1;
    2,   2;
   10,   7,   17;
   32,  75,   80,  230;
  271, 343, 1091, 1594, 4894;
  955, ...

M. I. Aroyo et al, Bilbao Crystallographic Server
International Union of Crystallography, International Tables for Crystallography, volumes A and E.
A. F. Palistrant, Complete scheme of four-dimensional crystallographic symmetry groups, Crystallography Reports, 57 (2012), 471-477.
W. Plesken and T. Schulz, CARAT Homepage
W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
B. Souvignier, The four-dimensional magnetic point and space groups, Z. Kristallogr., 221 (2006), 77-82.
Wikipedia: Space group, Crystallographic point group, Line group, Frieze group, Wallpaper group, Rod group, Layer group
Index entries for sequences related to groups

Cf. A006227, A293060, A293062, A293063.

A307290 Number of black-and-white colored (or proper magnetic) space groups in dimension n, counting enantiomorphs.

Original entry on oeis.org

0, 3, 46, 1191, 52439

Offset: 0

N. J. A. Sloane, Apr 08 2019

R. L. E. Schwarzenberger, Colour symmetry, Bulletin of the London Mathematical Society 16.3 (1984): 216-229.
B. Souvignier, Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6, Acta Cryst., A59 (2003), 210-220.

Cf. A006227, A307291, A307292.

a(n) = A307291(n) - 2 * A006227(n). - Andrey Zabolotskiy, Apr 09 2019

a(0), a(4) from Andrey Zabolotskiy, Apr 09 2019

A307291 Number of colored (or magnetic) space groups in dimension n, including enantiomorphs.

Original entry on oeis.org

2, 7, 80, 1651, 62227

Offset: 0

N. J. A. Sloane, Apr 08 2019

R. L. E. Schwarzenberger, Colour symmetry, Bulletin of the London Mathematical Society 16.3 (1984): 216-229.
B. Souvignier, Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6, Acta Cryst., A59 (2003), 210-220.

Cf. A307290.
Right border of A293063.
For non-magnetic space groups see A006227.

a(0), a(4) from Andrey Zabolotskiy, Apr 09 2019

A004031 Number of n-dimensional crystal systems.

Original entry on oeis.org

1, 1, 4, 7, 33, 59, 251

Offset: 0

N. J. A. Sloane

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).
(End)

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).

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.
W. Plesken and T. Schulz, CARAT Homepage
W. Plesken and T. Schulz, CARAT Homepage [Cached copy in pdf format (without subsidiary pages), with permission]
B. Souvignier, Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6, Acta Cryst., A59 (2003), 210-220.

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

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

A004032 Number of n-dimensional crystal families.

Original entry on oeis.org

1, 1, 4, 6, 23, 32, 91

Offset: 0

N. J. A. Sloane

H. Brown, R. Bülow, J. Neubüser, H. Wondratschek and H. Zassenhaus, Crystallographic Groups of Four-Dimensional Space. Wiley, NY, 1978, p. 52.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

W. Plesken and T. Schulz, Counting crystallographic groups in low dimensions, Experimental Mathematics, Volume 9 Issue 3 (2000), 321-478
Wikipedia, Space_group

Cf. A004027-A004031, A006226, A006227.

a(5)-a(6) from Henry Bottomley, Feb 03 2014

A383108 Number of crystallographic orbits in n dimensions, counting enantiomorphic pairs as distinct.

Original entry on oeis.org

1, 2, 30, 427

Offset: 0

Andrey Zabolotskiy, Apr 16 2025

Peter Engel, Geometric crystallography, D. Reidel Publishing Company, 1986. See Theorem 8.14 on p. 189.
Peter Engel, Geometric crystallography, in: P. M. Gruber and J. M. Wills, editors, Handbook of Convex Geometry, North-Holland, Amsterdam, 1993, Vol. B, pp. 989-1041. See Theorem 6.7 on p. 1027.

Online Dictionary of Crystallography, Crystallographic orbit.

Cf. A006227.

A092240 a(n) is the number of n-dimensional symmetry frieze designs (incorrect).

Original entry on oeis.org

7, 17, 230, 4783

Offset: 1

Nitsa Movshovitz-Hadar (nitsa(AT)tx.technion.ac.il), Oct 24 2004

dead

I suspect that some of the contributors to this entry have confused it with A004029. The term 4783 is probably wrong, since A004029(4) = 4783. - N. J. A. Sloane, Dec 27 2014

As far as I can tell, the values given in this sequence are not consistent with any possible interpretation of "Frieze". The standard Frieze groups are defined as the 2-D line groups (planar symmetry groups having a translation in one direction only). In one dimension, there are only 2 line groups (not 7), and 0 if we discount the groups having a translation in one direction (both of them). In three dimensions, there are the 219 or 230 crystallographic groups (depending on whether chiral copies are considered distinct), but these have translations in 3 directions. If we count groups having fewer than 3 translations, then there are just 80 layer groups (having translations in two directions), and 75 rod groups (having translations in one direction). - Brian Galebach, Oct 18 2016

			There are 7 strip patterns, i.e., 1-dimensional symmetry frieze designs; 17 wallpaper designs, i.e., 2-dimensional symmetry groups; 230 is the number of crystallographic groups, i.e., 3-dimensional symmetry designs; 4783 is the 4-dimensional extension of the above.

Piergiorgio Odifreddi, The Mathematical Century: The 30 Greatest Problems of the Last 100 Years, Princeton University Press, 2004, see p. 102.

Yanxi Liu, Collins, R.T., Tsin, Y., A computational model for periodic pattern perception based on frieze and wallpaper groups, IEEE Trans. Pattern Analysis and Machine Intelligence, 26 (2004), 354-371.

Cf. A004029, A006227.

cp's OEIS Frontend

A307289 Erroneous version of A006227.

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A004029 Number of n-dimensional space groups.

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A004027 Number of arithmetic n-dimensional crystal classes.

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A293061 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, counting enantiomorphs.

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A307290 Number of black-and-white colored (or proper magnetic) space groups in dimension n, counting enantiomorphs.

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A307291 Number of colored (or magnetic) space groups in dimension n, including enantiomorphs.

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A004031 Number of n-dimensional crystal systems.

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A004032 Number of n-dimensional crystal families.

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A383108 Number of crystallographic orbits in n dimensions, counting enantiomorphic pairs as distinct.

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A092240 a(n) is the number of n-dimensional symmetry frieze designs (incorrect).

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