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In geometry, the Cairo pentagonal tiling is a dual semiregular tiling of the Euclidean plane. It is given its name because several streets in Cairo are paved in this design.[1][2] It is one of 14 known isohedral pentagon tilings.

It is also called MacMahon's net[3] after Percy Alexander MacMahon and his 1921 publication New Mathematical Pastimes.[4]

Conway calls it a 4-fold pentille.[5]

These are not regular pentagons: their sides are not equal (they have four long ones and one short one in the ratio 1:sqrt(3)-1[6]), and their angles in sequence are 120°, 120°, 90°, 120°, 90°.

Relation to hexagonal tilings

This tiling can be seen as the union of two flattened perpendicular hexagonal tilings. Each hexagon is divided into four pentagons. The two hexagons can also be distorted to be concave, leading to concave pentagons.[7]

Dual tiling

It is the dual of the snub square tiling, made of two squares and three equilateral triangles around each vertex.[8]

Related polyhedra and tilings

As a dual to the snub square tiling the geometric proportions are fixed for this tiling. However it can be adjusted to other geometric forms with the same topological connectivity and different symmetry. For example, this rectangular tiling is topologically identical.

The Cairo pentagonal tiling is third in a series of dual snub polyhedra and tilings with face configuration V3.3.4.3.n.

Tilings of regular polygons
List of uniform tilings

Notes

Alsina, Claudi; Nelsen, Roger B. (2010), Charming proofs: a journey into elegant mathematics, Dolciani mathematical expositions 42, Mathematical Association of America, p. 164, ISBN 978-0-88385-348-1.
Martin, George Edward (1982), Transformation Geometry: An Introduction to Symmetry, Undergraduate Texts in Mathematics, Springer, p. 119, ISBN 978-0-387-90636-2.
O'Keeffe, M.; Hyde, B. G. (1980), "Plane nets in crystal chemistry", Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences 295 (1417): 553–618, doi:10.1098/rsta.1980.0150, JSTOR 36648.
Macmahon, Major P. A. (1921), New Mathematical Pastimes, University Press.
John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
http://catnaps.org/islamic/geometry2.html
Defining a cairo type tiling

Weisstein, Eric W., "Dual tessellation", MathWorld.

Grünbaum, Branko ; and Shephard, G. C. (1987). Tilings and Patterns. New York: W. H. Freeman. ISBN 0-7167-1193-1. (Chapter 2.1: Regular and uniform tilings, p.58-65) (Page 480, Tilings by polygons, #24 of 24 polygonal isohedral types by pentagons)

Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 38. ISBN 0-486-23729-X.
Wells, David, The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 23, 1991.