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A242941 a(n) = number of convex uniform tessellations in dimension n. 0
1, 11, 28, 143 (list; graph; refs; listen; history; text; internal format)



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


B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, 4 (1994), 49-56.

N. W. Johnson, Uniform Polytopes, Cambridge University Press, 1991.


Table of n, a(n) for n=1..4.

A. Andreini, Sulle reti di poliedri regolari e semiregolari e sulle correspondenti reto correlative., 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, Europ. J. Combinatorics, Volume 21, Issue 6 (2000), 807-814.

G. Olshevsky, Uniform Panoploid Tetracombs (2006)

Wikipedia, Convex uniform honeycomb


Cf. A068599.

List of coordination sequences for the 11 uniform 2D tilings: A008458(the planar net, A008486 (6^3), A008574 ( and, A008576 (4.8.8), A008579 (, A008706(, A072154 (4.6.12), A219529 (, A250120(, 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.

Sequence in context: A061952 A140677 A264443 * A018944 A061086 A201633

Adjacent sequences:  A242938 A242939 A242940 * A242942 A242943 A242944




Felix Fröhlich, May 27 2014


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



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Last modified February 16 14:47 EST 2019. Contains 320163 sequences. (Running on oeis4.)