A004027 -id:A004027

A263840

Erroneous version of A004027.

+20
0

1, 2, 13, 73, 710, 6079, 85311

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

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

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

KEYWORD

dead

STATUS

approved

A004029

Number of n-dimensional space groups.
(Formerly M2103)

+10
14

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

REFERENCES

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).

LINKS

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

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

CROSSREFS

Cf. A006227, A004027, A293060.

KEYWORD

nonn,hard,more,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A006226

Number of abstract n-dimensional crystallographic point groups.
(Formerly M1916)

+10
13

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

REFERENCES

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).

LINKS

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

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

CROSSREFS

Cf. A004027, A004028, A004029.

KEYWORD

hard,nonn,more,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A004028

Number of geometric n-dimensional crystal classes.
(Formerly M1965)

+10
9

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

REFERENCES

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).

LINKS

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

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.

CROSSREFS

Cf. A004027, A004029.

KEYWORD

nonn,hard,nice,more

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A006227

Number of n-dimensional space groups (including enantiomorphs).
(Formerly M2104)

+10
9

1, 2, 17, 230, 4894, 222097

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

COMMENTS

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

REFERENCES

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).

LINKS

Table of n, a(n) for n=0..5.

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

CROSSREFS

Cf. A004027, A004029, A293061.

KEYWORD

nonn,hard,more,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A004032

Number of n-dimensional crystal families.
(Formerly M3289)

+10
1

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

REFERENCES

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).

LINKS

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

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

Wikipedia, Space_group

CROSSREFS

Cf. A004027-A004031, A006226, A006227.

KEYWORD

hard,nonn,nice

AUTHOR

N. J. A. Sloane

EXTENSIONS

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

STATUS

approved

A059104

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

+10
1

1, 2, 10, 74, 1060, 38746

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

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

LINKS

Table of n, a(n) for n=1..6.

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]

CROSSREFS

Cf. A004027, A004028, A004029, A059105.

KEYWORD

nonn,hard,nice,more

AUTHOR

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

STATUS

approved

A059105

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

+10
1

0, 0, 1, 4, 101, 5004

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,4

LINKS

Table of n, a(n) for n=1..6.

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]

CROSSREFS

Cf. A004027, A004028, A004029, A059104.

KEYWORD

nonn,hard,nice,more

AUTHOR

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

STATUS

approved

A307288

Number of arithmetic classes of space groups in dimension n, including enantiomorphs.

+10
1

1, 2, 13, 73, 780, 6079

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,2

LINKS

Table of n, a(n) for n=0..5.

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

Wikipedia, Space group

CROSSREFS

For the case when enantiomorphs are not included see A004027.

KEYWORD

nonn,more

AUTHOR

N. J. A. Sloane, Apr 08 2019

STATUS

approved

A052487

Number of maximal irreducible integral matrix groups in n dimensions.

+10
0

1, 2, 3, 6, 7, 17, 7, 26, 20, 46, 9

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

REFERENCES

Peter Engel, Geometric Crystallography, D. Reidel, Dordrecht, Holland, 1986. ISBN 90-277-2339-7. See Table 6.3, p. 111.

LINKS

Table of n, a(n) for n=1..11.

J. H. Conway and N. J. A. Sloane, Low-Dimensional Lattices II: Subgroups of GL(n,Z), Proc. Royal Soc. London, A 419 (1988), 29-68, doi:10.1098/rspa.1988.0095.

Bernd Souvignier, Irreducible finite integral matrix groups of degree 8 and 10, Math. Comp. 63 (1994), 335-350.

CROSSREFS

Cf. A004027.

KEYWORD

nonn,more,nice

AUTHOR

Frederick G. Schmitt (fred(AT)marin.cc.ca.us), Mar 16 2000

EXTENSIONS

a(10) from Souvignier and a(11) from Conway & Sloane added by Andrey Zabolotskiy, Apr 17 2023

STATUS

approved