A293060 -id:A293060 - cp's OEIS frontend

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

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.

A293062 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional magnetic subperiodic groups in n-dimensional space, not counting enantiomorphs.

Original entry on oeis.org

2, 5, 7, 31, 31, 80, 122, 360, 528, 1594, 1025

Offset: 0

Andrey Zabolotskiy, Sep 29 2017

Magnetic groups are also known as antisymmetry groups, or black-white, or two-color crystallographic groups.
T(n,0) count n-dimensional magnetic crystallographic point groups, T(n,n) count n-dimensional magnetic space groups. The name "subperiodic groups" is usually related to the case 0 < k < n only, i.e., magnetic groups of n-dimensional objects including k independent translations which are subgroups of some n-dimensional magnetic space groups.
The Bohm-Koptsik symbols for these groups are G_{n,k}^1 (or G_{n+1,n,k}; the difference arises only when we consider enantiomorphism), except for the case k=n, when it is G_n^1 (or G_{n+1,n}).
T(2,1) are band groups.
T(3,3) are Shubnikov groups.
For T(n,0) and T(n,n), see [Souvignier, 2006, table 1]. See Litvin for the cases when there are no enantiomorphs: rows 1-2, T(3,2). For T(3,1), see, e.g., [Palistrant & Jablan, 1991].

			The triangle begins:
     2;
     5,   7;
    31,  31,  80;
   122, 360, 528, 1594;
  1025, ...

H. Grimmer, Comments on tables of magnetic space groups, Acta Cryst., A65 (2009), 145-155.
D. B. Litvin, Magnetic Group Tables
A. F. Palistrant and S. V. Jablan, Enantiomorphism of three-dimensional space and line multiple antisymmetry groups, Publications de l'Institut Mathématique, 49(63) (1991), 51-60.
B. Souvignier, The four-dimensional magnetic point and space groups, Z. Kristallogr., 221 (2006), 77-82.
Index entries for sequences related to groups

Cf. A293060, A293061, A293063.

T(n,n) = A293060(n+1,n).

A293063 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional magnetic subperiodic groups in n-dimensional space, counting enantiomorphs.

Original entry on oeis.org

2, 5, 7, 31, 31, 80, 122, 394, 528, 1651, 1202

Offset: 0

Andrey Zabolotskiy, Sep 29 2017

Magnetic groups are also known as antisymmetry groups, or black-white, or two-color crystallographic groups.
T(n,0) count n-dimensional magnetic crystallographic point groups, T(n,n) count n-dimensional magnetic space groups (A307291). The name "subperiodic groups" is usually related to the case 0 < k < n only, i.e., magnetic groups of n-dimensional objects including k independent translations which are subgroups of some n-dimensional magnetic space groups.
The Bohm-Koptsik symbols for these groups are G_{n,k}^1, except for the case k=n, when it is G_n^1.
T(2,1) are band groups.
T(3,3) are Shubnikov groups.
For T(n,0) and T(n,n), see [Souvignier, 2006, table 1]. For rows 1-3, see Litvin.

			The triangle begins:
     2;
     5,   7;
    31,  31,  80;
   122, 394, 528, 1651;
  1202, ...

H. Grimmer, Comments on tables of magnetic space groups, Acta Cryst., A65 (2009), 145-155.
D. B. Litvin, Magnetic Group Tables
B. Souvignier, The four-dimensional magnetic point and space groups, Z. Kristallogr., 221 (2006), 77-82.
Index entries for sequences related to groups

Cf. A293060, A293061, A293062, A307291.

cp's OEIS Frontend

A004029 Number of n-dimensional space groups.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Extensions

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.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

A293062 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional magnetic subperiodic groups in n-dimensional space, not counting enantiomorphs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A293063 Triangle read by rows (n >= 0, 0 <= k <= n): T(n,k) = number of k-dimensional magnetic subperiodic groups in n-dimensional space, counting enantiomorphs.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs