A006226 -id:A006226 - cp's OEIS frontend

A131936 Partial sums of A006226.

Original entry on oeis.org

1, 3, 12, 30, 148, 387, 1981

Offset: 1

Jonathan Vos Post, Oct 05 2007

nonn

Cf. A006226.

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

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

A004028 Number of geometric n-dimensional crystal classes.

Original entry on oeis.org

1, 2, 10, 32, 227, 955, 7103

Offset: 0

N. J. A. Sloane

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

Number of n-dimensional crystallographic point groups (not counting enantiomorphs). - Andrey Zabolotskiy, Jul 08 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.
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).

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.

Cf. A004027, A004029.

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.

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.

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

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.

cp's OEIS Frontend

A131936 Partial sums of A006226.

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A004029 Number of n-dimensional space groups.

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

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A004027 Number of arithmetic n-dimensional crystal classes.

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A004028 Number of geometric n-dimensional crystal classes.

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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.

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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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A004031 Number of n-dimensional crystal systems.

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A004032 Number of n-dimensional crystal families.

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

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