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

%I A242941 #90 Feb 24 2019 12:49:45
%S A242941 1,11,28,143
%N A242941 a(n) is the number of convex uniform tessellations in dimension n.
%C A242941 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.
%C A242941 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
%C A242941 From _Felix Fröhlich_, Feb 03 2019: (Start)
%C A242941 The 11 convex uniform tilings are all illustrated in Kepler, 1619. For an argument that exactly 11 such tilings exist, see Grünbaum, Shephard, 1977.
%C A242941 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)
%D A242941 H. S. M. Coxeter, Regular Polytopes, Third Edition, Dover Publications, 1973, ISBN 9780486614809.
%D A242941 B. Grünbaum, Uniform tilings of 3-space, Geombinatorics, Vol. 4, No. 2 (1994), 49-56.
%D A242941 N. W. Johnson, Uniform Polytopes, [To appear, cf. Weiss, Stehle, 2017].
%H A242941 A. Andreini, <a href="/A242941/a242941.pdf">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]</a>, Mem. Società Italiana della Scienze, Ser.3, 14 (1905), 75-129.
%H A242941 M. Deza and M. Shtogrin, <a href="https://doi.org/10.1006/eujc.1999.0385">Uniform Partitions of 3-space, their Relatives and Embedding</a>, European Journal of Combinatorics, Vol. 21, No. 6 (2000), 807-814.
%H A242941 B. Grünbaum and G. C. Shephard, <a href="https://doi.org/10.1080/0025570X.1977.11976655">Tilings by Regular Polygons</a>, Mathematics Magazine, Vol. 50, No. 5 (1977), 227-247.
%H A242941 J. Kepler, <a href="https://archive.org/details/ioanniskepplerih00kepl/page/n73">Harmonices Mundi [The Harmony of the World]</a> (1619).
%H A242941 G. Olshevsky, <a href="http://bendwavy.org/4HONEYS.pdf">Uniform Panoploid Tetracombs</a> (2006)
%H A242941 A. I. Weiss and E. M. Stehle, <a href="https://doi.org/10.26493/2590-9770.1231.403">Norman W. Johnson (12 November 1930 to 13 July 2017)</a>, The Art of Discrete and Applied Mathematics, Vol. 1, No. 1 (2018).
%H A242941 Wikipedia, <a href="https://en.wikipedia.org/wiki/Convex_uniform_honeycomb">Convex uniform honeycomb</a>
%H A242941 Wikipedia, <a href="https://en.wikipedia.org/wiki/List_of_convex_uniform_tilings">List of convex uniform tilings</a>
%Y A242941 Cf. A068599.
%Y A242941 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).
%Y A242941 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.
%K A242941 nonn,nice,hard,more
%O A242941 1,2
%A A242941 _Felix Fröhlich_, May 27 2014
%E A242941 Edited by _N. J. A. Sloane_, Feb 15 2018
%E A242941 Edited by _Felix Fröhlich_, Feb 03-10 2019