A068599 -id:A068599 - cp's OEIS frontend

A068600 Number of n-uniform tilings having n different arrangements of polygons about their vertices.

11, 20, 39, 33, 15, 10, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Brian Galebach, Mar 28 2002

nonn

Sequence gives the number of edge-to-edge regular-polygon tilings having n topologically distinct vertex types, with each vertex type having a different arrangement of surrounding polygons. Does not allow for tilings with two or more vertex types having the same arrangement of surrounding polygons, even when those vertices are topologically distinct. There are no 8- or higher-uniform tilings having the equivalent number of distinct polygon arrangements.

There are eleven 1-uniform tilings (also called the "Archimedean" tessellations) which comprise the three regular tessellations (all triangles, squares, or hexagons) plus the eight semiregular tessellations. (See A250120. - N. J. A. Sloane, Nov 29 2014)

This sequence was originally calculated by Otto Krotenheerdt.
Branko Grünbaum and G. C. Shephard, Tilings and Patterns. W. H. Freeman, New York, 1987, page 69.
Krotenheerdt, Otto. "Die homogenen Mosaike n-ter Ordnung in der euklidischen Ebene," Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg. Math.-natur. Reihe, 18(1969), 273-290; 19 (1970)19-38 and 97-122.

Steven Dutch, Uniform Tilings
Brian L. Galebach, n-Uniform Tilings
Ng Lay Ling, Honours Project - Tilings and Patterns.

Cf. A068599.

List of coordination sequences for uniform planar nets: A008458 (the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706 (3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120(3.3.3.3.6), A250122 (3.12.12).

A268184 Number of n-isohedral edge-to-edge tilings of regular polygons.

Original entry on oeis.org

3, 13, 29, 70, 140, 267, 559

Offset: 1

Brian Galebach, Jan 28 2016

An n-isohedral tiling has n transitivity classes (or "orbits") of faces with respect to the symmetry group of the tiling.

			The three 1-isohedral tilings are the regular tilings (triangles, squares, hexagons). Of the 13 2-isohedral tilings, there are three with triangles and squares, eight with triangles and hexagons, one with triangles and dodecagons, and one with squares and octagons.

D. Chavey, Periodic Tilings and Tilings by Regular Polygons I, Thesis, 1984, pp. 165-172 gives the 2-isohedral edge-to-edge tilings of regular polygons.
D. Chavey, Tiling by Regular Polygons II: A Catalog of Tilings, Computers & Mathematics with Applications, Volume 17, Issues 1-3, 1989, Pages 147-165, illustrates 27 of the 29 3-isohedral edge-to-edge tilings of regular polygons, but classifies one (3^3.4^2; 3^2.4.3.4)2 on page 152 as 6-isohedral.
Brian Galebach, Announcement of 7-Isohedral Tiling Count, Facebook

Analogous to the n-uniform edge-to-edge tilings, which has n orbits of vertices, as opposed to faces (A068599).

a(7) from Brian Galebach, Dec 23 2016

A299780 Triangle read by rows: T(n,m) = number of n-uniform tilings having m different arrangements of polygons about their vertices, n >= 1 and 1 <= m <= n.

Original entry on oeis.org

11, 0, 20, 0, 22, 39, 0, 33, 85, 33, 0, 74, 149, 94, 15, 0, 100, 284, 187, 92, 10, 0

Offset: 1

Omar E. Pol, Mar 30 2018

Taken from Brian Galebach's square array (see link).

			Triangle begins:
  11;
   0,  20;
   0,  22,  39;
   0,  33,  85,  33;
   0,  74, 149,  94, 15;
   0, 100, 284, 187, 92, 10;
...
Other known positive terms are T(7,7) = 7, T(8,7) = 20, T(9,8) = 8, T(10,8) = 27 and T(11,9) = 1.

Brian L. Galebach, n-Uniform Tilings
José Ezequiel Soto Sánchez, Asla Medeiros e Sá, Luiz Henrique de Figueiredo, Acquiring periodic tilings of regular polygons from images, The Visual Computer (2019) Vol. 35, Issue 6-8, 899-907.

Row sums gives A068599.
Leading diagonal is A068600.
Column 1 gives 11 together with A000004.
Cf. A299781, A299782.

A299781 Triangle read by rows: T(n,m) = number of k-uniform tilings having m different arrangements of polygons about their vertices, for k = 1..n, with 1 <= m <= n.

Original entry on oeis.org

11, 11, 20, 11, 42, 39, 11, 75, 124, 33, 11, 149, 273, 127, 15, 11, 249, 557, 314, 107, 10, 11

Offset: 1

Omar E. Pol, Mar 30 2018

Column m lists the partial sums of the m-th column of triangle A299780.

			Triangle begins:
  11;
  11,  20;
  11,  42,  39;
  11,  75, 124,  33;
  11, 149, 273, 127,  15;
  11, 249, 557, 314, 107, 10;
...

Brian L. Galebach, n-Uniform Tilings

Column 1 gives A010850.
Leading diagonal is A068600.
Row sums give A299782.
Cf. A068599, A299780.

A299782 a(n) is the total number of k-uniform tilings, for k = 1..n.

Original entry on oeis.org

11, 31, 92, 243, 575, 1248

Offset: 1

Omar E. Pol, Mar 30 2018

			For n = 3 there are 11 uniform tilings, 20 2-uniform tilings and 61 3-uniform tilings. The sum of them is 11 + 20 + 61 = 92, so (3) = 92.

Brian L. Galebach, n-Uniform Tilings

Partial sums of A068599.
Row sums of A299781.
Cf. A068600, A299780.

A242941 a(n) is the number of convex uniform tessellations in dimension n.

Original entry on oeis.org

1, 11, 28, 143

Offset: 1

Felix Fröhlich, May 27 2014

Terms for n > 4 have not been determined so far. Alfredo Andreini in 1905 gave a value of 25 for a(3), later found to be incorrect. The value 28 for a(3) was given by Norman Johnson in 1991 and later in 1994 independently by Branko Grünbaum. The value for a(4) was given by George Olshevsky in 2006.
Deza and Shtogrin (2000) agree that the value of a(3) is 28, although the authors do not provide a proof. - Felix Fröhlich, Nov 29 2014
From Felix Fröhlich, Feb 03 2019: (Start)
The 11 convex uniform tilings are all illustrated in Kepler, 1619. For an argument that exactly 11 such tilings exist, see Grünbaum, Shephard, 1977.
In dimension 2, the definition of "uniform polytope" usually seems to be equivalent to the regular polygons in order to exclude polygons that alternate two different edge-lengths. Applying this principle retroactively to dimension 1 (as done, as I assume, by Coxeter, see Coxeter, 1973, p. 129) yields a(1) = 1. (End)

H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486614809.
B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, Vol. 4, No. 2 (1994), 49-56.
N. W. Johnson, Uniform Polytopes, [To appear, cf. Weiss, Stehle, 2017].

A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle correspondenti reto correlative [On the regular and semiregular nets of polyhedra and on the corresponding correlative nets], Mem. Società Italiana della Scienze, Ser.3, 14 (1905), 75-129.
M. Deza and M. Shtogrin, Uniform Partitions of 3-space, their Relatives and Embedding, European Journal of Combinatorics, Vol. 21, No. 6 (2000), 807-814.
B. Grünbaum and G. C. Shephard, Tilings by Regular Polygons, Mathematics Magazine, Vol. 50, No. 5 (1977), 227-247.
J. Kepler, Harmonices Mundi [The Harmony of the World] (1619).
G. Olshevsky, Uniform Panoploid Tetracombs (2006)
A. I. Weiss and E. M. Stehle, Norman W. Johnson (12 November 1930 to 13 July 2017), The Art of Discrete and Applied Mathematics, Vol. 1, No. 1 (2018).
Wikipedia, Convex uniform honeycomb
Wikipedia, List of convex uniform tilings

Cf. A068599.
List of coordination sequences for the 11 uniform 2D tilings: A008458(the planar net 3.3.3.3.3.3), A008486 (6^3), A008574 (4.4.4.4 and 3.4.6.4), A008576 (4.8.8), A008579 (3.6.3.6), A008706(3.3.3.4.4), A072154 (4.6.12), A219529 (3.3.4.3.4), A250120(3.3.3.3.6), A250122 (3.12.12).
List of coordination sequences for the 28 uniform 3D tilings: cab: A299266, A299267; crs: A299268, A299269; fcu: A005901, A005902; fee: A299259, A299265; flu-e: A299272, A299273; fst: A299258, A299264; hal: A299274, A299275; hcp: A007899, A007202; hex: A005897, A005898; kag: A299256, A299262; lta: A008137, A299276; pcu: A005899, A001845; pcu-i: A299277, A299278; reo: A299279, A299280; reo-e:  A299281, A299282; rho: A008137, A299276; sod: A005893, A005894; sve: A299255, A299261; svh: A299283, A299284; svj: A299254, A299260; svk: A010001, A063489; tca: A299285, A299286; tcd: A299287, A299288; tfs: A005899, A001845; tsi: A299289, A299290; ttw: A299257, A299263; ubt: A299291, A299292; bnn: A007899, A007202. See the Proserpio link in A299266 for overview.

Edited by N. J. A. Sloane, Feb 15 2018

Edited by Felix Fröhlich, Feb 03-10 2019

cp's OEIS Frontend

A068600 Number of n-uniform tilings having n different arrangements of polygons about their vertices.

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A268184 Number of n-isohedral edge-to-edge tilings of regular polygons.

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A299780 Triangle read by rows: T(n,m) = number of n-uniform tilings having m different arrangements of polygons about their vertices, n >= 1 and 1 <= m <= n.

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A299781 Triangle read by rows: T(n,m) = number of k-uniform tilings having m different arrangements of polygons about their vertices, for k = 1..n, with 1 <= m <= n.

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A299782 a(n) is the total number of k-uniform tilings, for k = 1..n.

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A242941 a(n) is the number of convex uniform tessellations in dimension n.

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