A004029 -id:A004029 - cp's OEIS frontend

A256413 Number of n-dimensional Bravais lattices.

Original entry on oeis.org

1, 1, 5, 14, 64, 189, 841

Offset: 0

N. J. A. Sloane, Apr 04 2015

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.
Lomont, J. S. "Crystallographic Point Groups." 4.4 in Applications of Finite Groups. New York: Dover, pp. 132-146, 1993.
Yale, P. B. "Crystallographic Point Groups." 3.4 in Geometry and Symmetry. New York: Dover, pp. 103-108, 1988.

D. Freittloh, Highly symmetric fundamental cells for lattices in R^2 and R^3, arXiv.1305.1798 [math.CO], 2013.
S. J. Heyes, Illustration of the 14 possible 3-D Bravais lattices from Lecture 1. Fundamental Aspects of Solids & Sphere Packing. - Analysing a 3D solid
Opgenorth, J; Plesken, W; Schulz, T, Crystallographic Algorithms and Tables, Acta Crystallogr. A, 54 (1998), 517-531.
Pegg, Ed Jr., Bravais Lattice.
W. Plesken and W. Hanrath, The lattices of six-dimensional Euclidean space, Math. Comp., 43 (1984), 573-587. [Warning: gives wrong value for a(6).]
B. Souvignier, Enantiomorphism of Crystallographic Groups in Higher Dimensions with Results in Dimensions Up to 6, Acta Cryst. A 59, 210-220, 2003.
Bernd Souvignier, Space groups, 2007, p. 30

Cf. A004029.

A004030 is an incorrect version found in the literature.

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.

A166742 Arises in enumeration of three-dimensional crystallographic Seifert and co-Seifert fibrations.

Original entry on oeis.org

1, 4, 7, 9, 19, 29, 33, 76, 144, 169

Offset: 1

Jonathan Vos Post, Oct 21 2009

p.25 of Ratcliffe: "The information in Table 1 was obtained by computer calculations. Finally, the 10 closed flat space forms in Table 1 have IT numbers 1,4,7,9,19,29,33,76,144,169."

The "IT numbers" here are merely numbers of certain 3D space groups in the International Tables for Crystallography. Thus, the terms of this sequence don't have their own mathematical meaning. - Andrey Zabolotskiy, Jul 05 2017

John G. Ratcliffe and Steven T. Tschantz, Fibered orbifolds and crystallographic groups, arXiv:0804.0427 [math.GT], 2008-2009.

Cf. A004029.

A307293 Number of color plane groups of index n.

Original entry on oeis.org

17, 46, 23, 96, 14, 90, 15, 166, 40, 75, 13, 219, 16, 80, 34, 262, 14, 174, 15, 205, 38, 88, 13, 433, 31, 103, 48, 222, 14, 213, 15, 395, 36, 111, 24, 452, 16, 116, 40, 416, 14, 250, 15, 265, 62, 124, 13, 741, 32, 193, 38, 300, 14, 303, 24, 468, 42, 147, 13, 627, 16

Offset: 1

N. J. A. Sloane, Apr 08 2019

nonn

			a(1) = A004029(2), a(2) = A307292(2).

Branko Grünbaum and G. C. Shephard, Tilings and Patterns, W. H. Freeman, New York, 1987, chapter 8 "Colored patterns and tilings".
Thomas W. Wieting, The Mathematical Theory of Chromatic Plane Ornaments, Marcel Dekker, Inc., 1982. See Table 11 at pages 250-254.

R. L. E. Schwarzenberger, Colour symmetry: Part two, Bulletin of the London Mathematical Society 16.3 (1984): 216-229. See page 221.
R. L. E. Schwarzenberger, Review of "The Mathematical Theory of Chromatic Plane Ornaments" by Thomas W. Wieting, Bull. London Math. Soc., 15 (1983), 163-164.
Marjorie Senechal, Color groups, Discrete Applied Mathematics, 1 (1979), 51-73.

Cf. A004029, A307292.

a(p) = 16, 15, 14, 13 if p == 1, 7, 5, 11 (mod 12), respectively, and a(p^2) = a(p) + 17, where p is a prime greater than 3. These formulas are found by Senechal. Schwarzenberger (1983) says that her results are correct for these cases, while some other results have essentially been corrected by Wieting. - Andrey Zabolotskiy, May 18 2022

a(1) and a(16)-a(60) from Wieting added by Andrey Zabolotskiy, Apr 09 2019

a(61) added by Andrey Zabolotskiy, May 18 2022

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

A256413 Number of n-dimensional Bravais lattices.

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

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A166742 Arises in enumeration of three-dimensional crystallographic Seifert and co-Seifert fibrations.

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A307293 Number of color plane groups of index n.

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

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