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

1, 11, 28, 143

Offset: 1

Author

Felix Fröhlich, May 27 2014

Keywords

• nonn
• nice
• hard
• more

Comments

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)

References

• 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].

Links

• 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

Crossrefs

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.

Extensions

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

Edited by Felix Fröhlich, Feb 03-10 2019