A004028 -id:A004028 - cp's OEIS frontend

A263839 Erroneous version of A004028.

Original entry on oeis.org

1, 2, 10, 32, 227, 955, 7104

Offset: 0

dead

W. Plesken and T. Schulz, Counting crystallographic groups in low dimensions, Experimental Mathematics, 9 (No. 3, 2000), 407-411.

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

A006226 Number of abstract n-dimensional crystallographic point groups.

Original entry on oeis.org

1, 2, 9, 18, 118, 239, 1594

Offset: 0

N. J. A. Sloane

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.
T. Janssen, Crystallographic Groups. North-Holland, Amsterdam, 1973, p. 73.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

A. C. Hurley, Finite rotation groups and crystal classes in four dimensions, Proc. Cambridge Philos. Soc. 47, (1951). 650-661.
A. C. Hurley, Crystal Classes of Four-Dimensional Space R4, Acta Crystallog., 22 (1967), see especially p. 605.
W. Plesken and T. Schulz, Counting crystallographic groups in low dimensions, Experimental Mathematics, 9 (No. 3, 2000), 407-411.
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]
E. S. Rosenthal & N. J. A. Sloane, Correspondence, 1975

Cf. A004027, A004028, A004029.

Two more terms from W. Plesken and T. Schulz (tilman(AT)momo.math.rwth-aachen.de), Feb 27 2001

Offset corrected by Andrey Zabolotskiy, Jul 10 2017

A006227 Number of n-dimensional space groups (including enantiomorphs).

Original entry on oeis.org

1, 2, 17, 230, 4894, 222097

Offset: 0

N. J. A. Sloane

Right border of A293061. - Andrey Zabolotskiy, Oct 07 2017

Colin Adams, The Tiling Book, AMS, 2022; see p. 59.
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.
T. Janssen, Crystallographic Groups. North-Holland, Amsterdam, 1973, p. 119.
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
Manuel Caroli and Monique Teillaud. Delaunay triangulations of closed Euclidean dorbifolds. Discrete and Computational Geometry, Springer Verlag, 2016, 55 (4), pp.827-853. 10.1007/s00454-016-9782-6, hal-01294409
J. Neubüser, B. Souvignier and H. Wondratschek, Corrections to Crystallographic Groups of Four-Dimensional Space by Brown et al. (1978) [New York: Wiley and Sons], Acta Cryst., A58 (2002), 301.
J. Opgenorth, W. Plesken and T. Schulz, Crystallographic Algorithms and Tables, Acta Cryst., A54 (1998), 517-531.
W. Plesken, J. Opgenorth and T. Schulz, CARAT - a package for mathematical crystallography, Journal of Applied Crystallography, 31 (1998), 827-828.
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.
E. S. Rosenthal & N. J. A. Sloane, Correspondence, 1975
B. Souvignier, Enantiomorphism of crystallographic groups in higher dimensions with results in dimensions up to 6, Acta Cryst., A59 (2003), 210-220.
The Fascination of Crystals and Symmetry, 230 (The space group list project)
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. A004027, A004029, A293061.

a(4) corrected according to Neubüser, Souvignier and Wondratschek (2002) - Susanne Wienand, May 19 2014

a(5) added according to Souvignier (2003); a(6) should not be extracted from that paper because it uses the old incorrect CARAT data for d=6 - Andrey Zabolotskiy, May 19 2015

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

A293060 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, not counting enantiomorphs.

Original entry on oeis.org

1, 2, 2, 10, 7, 17, 32, 67, 80, 219, 227, 343, 1076, 1594, 4783, 955

Offset: 0

Andrey Zabolotskiy, Sep 29 2017

T(n,0) count n-dimensional crystallographic point groups (i.e., left border is A004028), T(n,n) count n-dimensional space groups (i.e., right border is A004029). 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
See [Palistrant, 2012, p. 476] for row 4.

			The triangle begins:
    1;
    2,   2;
   10,   7,   17;
   32,  67,   80,  219;
  227, 343, 1076, 1594, 4783;
  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. A004028, A004029, A293061, A293062, A293063.

A059104 Number of Bieberbach groups in dimension n: torsion-free crystallographic groups.

Original entry on oeis.org

1, 2, 10, 74, 1060, 38746

Offset: 1

Tilman Schulz (tilman(AT)momo.math.rwth-aachen.de), Feb 13 2001

Charlap, Leonard S., Bieberbach Groups and Flat Manifolds. Universitext. Springer-Verlag, New York, 1986. xiv+242 pp. ISBN: 0-387-96395-2 MR0862114 (88j:57042). See p. 6.
C. Cid, T. Schulz: Computation of Five and Six Dimensional Bieberbach Groups, Experimental Mathematics 10:1 (2001), 109-115
Manuel Caroli, Monique Teillaud. Delaunay triangulations of closed Euclidean dorbifolds. Discrete and Computational Geometry, Springer Verlag, 2016, 55 (4), pp.827-853. 10.1007/s00454-016-9782-6, hal-01294409; https://hal.inria.fr/hal-01294409/document

W. Plesken and T. Schulz, The 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]

Cf. A004027, A004028, A004029, A059105.

A059105 Number of n-dimensional torsion-free crystallographic groups with trivial center.

Original entry on oeis.org

0, 0, 1, 4, 101, 5004

Offset: 1

Tilman Schulz (tilman(AT)momo.math.rwth-aachen.de), Feb 13 2001

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]

Cf. A004027, A004028, A004029, A059104.

A381103 Number of permissible general positions in three-dimensional space groups obeying the crystallographic restriction theorem.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 96, 192

Offset: 1

Ambarneil Saha, Apr 14 2025

We can subdivide the 230 crystallographically permissible 3D space groups into 16 subsets based on the number of general positions (i.e., coordinate triplets whose values describe points occupied by symmetry-equivalent atoms in 3D space) specified by the symmetry operators in each subset. These numbers range from 1 (corresponding to exclusively one primitive triclinic space group, P1) to 192 (corresponding to the four face-centered cubic space groups Fm-3m, Fm-3c, Fd-3m, and Fd-3c). Multiplicities 1 and 9 (corresponding to exclusively one rhombohedral space group, R3h) represent the smallest subsets, whereas the largest subset is formed by the 63 space groups with multiplicity 8.

B. Kahr, A periodic-like table of space groups, Acta Cryst. E79, 124-128 (2023).
B. Souvignier, H. Wondratschek, M. I. Aroyo, G. Chapuis, and A. M. Glazer, International Tables for Crystallography Vol. A, Chapter 1.4, pp. 42-73 (2015). [alternative link]

Cf. A323383 (analog for the wallpaper groups).

Cf. A004028, A004029, A006227.

cp's OEIS Frontend

A263839 Erroneous version of A004028.

Views

Author

Keywords

References

A004029 Number of n-dimensional space groups.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A006226 Number of abstract n-dimensional crystallographic point groups.

Views

Author

Keywords

References

Links

Crossrefs

Extensions

A006227 Number of n-dimensional space groups (including enantiomorphs).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A004027 Number of arithmetic n-dimensional crystal classes.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

A293060 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional subperiodic groups in n-dimensional space, not counting enantiomorphs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A059104 Number of Bieberbach groups in dimension n: torsion-free crystallographic groups.

Views

Author

Keywords

References

Links

Crossrefs

A059105 Number of n-dimensional torsion-free crystallographic groups with trivial center.

Views

Author

Keywords

Links

Crossrefs

A381103 Number of permissible general positions in three-dimensional space groups obeying the crystallographic restriction theorem.

Views

Author

Keywords

Comments

Links

Crossrefs