In geometry, a Cairo pentagonal tiling is a tessellation of the Euclidean plane by congruent convex pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in Cairo.

It has also been studied as a crystal structure and appears in the art of M. C. Escher.

[4] Infinitely many different pentagons can form Cairo tilings, all with the same pattern of adjacencies between tiles and with the same decomposition into hexagons, but with varying edge lengths, angles, and symmetries.

In their tilings, the vertices with complementary angles alternate around each degree-four vertex.

One can also define analogous coordination sequences for the vertices of the tiling instead of for its tiles, but because there are two types of vertices (of degree three and degree four) there are two different coordination sequences arising in this way.

This form of the Cairo tiling inherits the property of the tilings by regular hexagons (unchanged by the flattening), that every edge is collinear with infinitely many other edges.

There is a unique equilateral pentagon that can form a type 4 Cairo tiling; it has five equal sides but its angles are unequal, and its tiling is bilaterally symmetric.

[4][13] Infinitely many other equilateral pentagons can form type 2 Cairo tilings.

[18][19] As of 2019 this pattern can still be seen as a surface decoration for square tiles near the Qasr El Nil Bridge and the El Behoos Metro station; other versions of the tiling are visible elsewhere in the city.

[20] Some authors including Martin Gardner have written that this pattern is used more widely in Islamic architecture, and although this claim appears to have been based on a misunderstanding, patterns resembling the Cairo tiling are visible on the 17th-century Tomb of I'timād-ud-Daulah in India, and the Cairo tiling itself has been found on a 17th-century Mughal jali.

[16] One of the earliest publications on the Cairo tiling as a decorative pattern occurs in a book on textile design from 1906.

[21] Inventor H. C. Moore filed a US patent on tiles forming this pattern in 1908.

[22] At roughly the same time, Villeroy & Boch created a line of ceramic floor tiles in the Cairo tiling pattern, used in the foyer of the Laeiszhalle in Hamburg, Germany.

The Cairo tiling has been used as a decorative pattern in many recent architectural designs; for instance, the city center of Hørsholm, Denmark, is paved with this pattern, and the Centar Zamet, a sports hall in Croatia, uses it both for its exterior walls and its paving tiles.

In the penta-graphene structure, the edges of the tiling incident to degree-four vertices form single bonds, while the remaining edges form double bonds.

[26] The Cairo tiling has been described as one of M. C. Escher's "favorite geometric patterns".

An image of this tessellation has also been used as the cover art for the 1974 first edition of H. S. M. Coxeter's book Regular Complex Polytopes.