# Convex pentagons that admit *i*-block transitive tilings

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## Abstract

The problem of classifying the convex pentagons that admit tilings of the plane is a long-standing unsolved problem. Previous to this article, there were 14 known distinct kinds of convex pentagons that admit tilings of the plane. Five of these types admit tile-transitive tilings (i.e. there is a single transitivity class with respect to the symmetry group of the tiling). The remaining 9 types do not admit tile-transitive tilings, but do admit either 2-block transitive tilings or 3-block transitive tilings; these are tilings comprised of clusters of 2 or 3 pentagons such that these clusters form tile-2-transitive or tile-3-transitive tilings. In this article, we present some combinatorial results concerning pentagons that admit *i*-block transitive tilings for \(i \in \mathbb {N}\). These results form the basis for an automated approach to finding all pentagons that admit *i*-block transitive tilings for each \(i \in \mathbb {N}\). We will present the methods of this algorithm and the results of the computer searches so far, which includes a complete classification of all pentagons admitting *i*-block transitive tilings for \(i \le 4\), among which is a new 15th type of convex pentagon that admits a tile-3-transitive tiling.

## Keywords

Tiling Tessellation Pentagon## Mathematics Subject Classification

05B45 52C20## Supplementary material

## References

- 1.Bagina, O.: Tiling the plane with congruent equilateral convex pentagons. J. Comb. Theory Ser. A
**105**(2), 221–232 (2004). doi: 10.1016/j.jcta.2003.11.002 MathSciNetCrossRefzbMATHGoogle Scholar - 2.Bagina, O.G.: Convex pentagons that tile the plane (types: 11112, 11122). Sib. Èlektron. Mat. Izv.
**9**, 478–530 (2012)MathSciNetzbMATHGoogle Scholar - 3.Gardner, M.: Time Travel and Other Mathematical Bewilderments. W. H. Freeman and Company, New York (1988)zbMATHGoogle Scholar
- 4.Grünbaum, B., Shephard, G.C.: The eighty-one types of isohedral tilings in the plane. Math. Proc. Camb. Philos. Soc.
**82**(2), 177–196 (1977)MathSciNetCrossRefzbMATHGoogle Scholar - 5.Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W. H. Freeman and Company, New York (1987)zbMATHGoogle Scholar
- 6.Heesch, H.: Aufbau der ebene aus kongruenten bereichen. Nachr. Ges. Wiss. Göttingen New Ser.
**1**, 115–117 (1935)zbMATHGoogle Scholar - 7.Heesch, H., Kienzle, O.: Flächenschluss. System der Formen lückenlos aneinanderschliessender Flachteile. Springer, Berlin (1963)Google Scholar
- 8.Hirschhorn, M.D., Hunt, D.C.: Equilateral convex pentagons which tile the plane. J. Comb. Theory Ser. A
**39**(1), 1–18 (1985). doi: 10.1016/0097-3165(85)90078-0 MathSciNetCrossRefzbMATHGoogle Scholar - 9.Kershner, R.B.: On paving the plane. Am. Math. Mon.
**75**, 839–844 (1968)MathSciNetCrossRefzbMATHGoogle Scholar - 10.Niven, I.: Convex polygons that cannot tile the plane. Am. Math. Mon.
**85**(10), 785–792 (1978). doi: 10.2307/2320624 MathSciNetCrossRefzbMATHGoogle Scholar - 11.Reinhardt, K.: Über die zerlegung der ebene in polygone. Ph.D. thesis, Univ. Frankfurt a.M. Noske (1918)Google Scholar
- 12.Reinhardt, K.: Über die zerlegung der hyperbolischen ebene in konvexe polygone. Jahresberichte der Deutschen Mathematiker-Vereinigung
**37**, 330–332 (1928)zbMATHGoogle Scholar - 13.Schattschneider, D.: Tiling the plane with congruent pentagons. Math. Mag.
**51**(1), 29–44 (1978)MathSciNetCrossRefzbMATHGoogle Scholar