A068599 Number of n-uniform tilings.

11, 20, 61, 151, 332, 673, 1472, 2850, 5960, 11866, 24459, 49794, 103082

Offset: 1

Author

Brian Galebach, Mar 28 2002

Keywords

• hard
• nice
• more
• nonn

Comments

Sequence gives the number of edge-to-edge regular-polygon tilings having n vertex classes relative to the symmetry of the tiling. Allows tilings with two or more vertex classes having the same arrangement of surrounding polygons (vertex type), as long as those classes are distinct within the symmetry of the tiling .

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.

References

• Marek Čtrnáct, Postings to Tiling Mailing List, 2021 (a(13) announced in posting on Dec 21 2021).
• B. Grünbaum and G. C. Shephard, Tilings and Patterns, an Introduction, Freeman, 1989; Exercise *6 on p. 70. See Sections 2.1 and 2.2.

Links

• D. P. Chavey, Periodic tilings and tilings by regular polygons, PhD thesis, Univ of Wisconsin, Madison, 1984 (gives a(3)).
• Marek Čtrnáct and Eryk Kopczyński, Tesselation catalog
• Steven Dutch, Uniform Tilings
• Brian Galebach, n-Uniform Tilings
• Brian Galebach, 7-Uniform Tiling Example, shows a tiling with 7 vertex classes (7-uniform), and 6 vertex types (6-Archimedean).
• El Jj, Deux (deux ?) minutes pour... classer les pavages !, youtube video (in French).
• Joris Kattemölle, Edge coloring lattice graphs, arXiv:2402.08752 [quant-ph], 2024.
• Ng Lay Ling, Honours Project - Tilings and Patterns.
• N. J. A. Sloane, Coordination Sequences, Planing Numbers, and Other Recent Sequences (II), Experimental Mathematics Seminar, Rutgers University, Jan 31 2019, Part I, Part 2, Slides. (Mentions this sequence)
• Hui Wang, Mengman Liu, Chuhua Ding, and Yi Ding, A systematic method of generating hinged tessellations by adding hinged plates based on dual graphs, Frontiers Archit. Res. (2024). See Table 1.
• Eric Weisstein's World of Mathematics, Uniform Tessellation

Crossrefs

Cf. A068600.

Extensions

151 and 332 found by Brian Galebach on Apr 30 2002, 673 on Aug 06 2003, 1472 on Apr 28 2020

a(8)-a(13) found by Marek Čtrnáct in 2021. - N. J. A. Sloane, Dec 21 2021