Cairo pentagonal tiling

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Cairo pentagonal tiling
1-uniform 9 dual.svg
TypeDual semiregular tiling
Facesirregular pentagons
Coxeter diagramCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png
Symmetry groupp4g, [4+,4], (4*2)
p4, [4,4]+, (442)
Rotation groupp4, [4,4]+, (442)
Dual polyhedronSnub square tiling
Face configurationV3.
Tiling face 3-3-4-3-4.svg

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 15 known monohedral 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]

As a 2-dimensional crystal net, it shares a special feature with the honeycomb net. Both nets are examples of standard realization, the notion introduced by M. Kotani and T. Sunada for general crystal nets.[6][7]


Geometry of each pentagon

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[8]), and their angles in sequence are 120°, 120°, 90°, 120°, 90°, it is represented by with face configuration V3.

It is similar to the prismatic pentagonal tiling with face configuration V3., which has its right angles adjacent to each other.


The Cairo pentagonal tiling has two lower symmetry forms given as monohedral pentagonal tilings types 4 and 8:

p4 (442) pgg (22×)
P5-type4.png P5-type8.png
Prototile p5-type4.png
b=c, d=e
Prototile p5-type8.png
Lattice p5-type4.png Lattice p5-type8.png

Dual tiling[edit]

It is the dual of the snub square tiling, made of two squares and three equilateral triangles around each vertex.[9]

P2 dual.png

Relation to hexagonal tilings[edit]

This tiling can be seen as the union of two perpendicular hexagonal tilings, flattened by a ratio of . Each hexagon is divided into four pentagons; the two hexagons can also be distorted to be concave, leading to concave pentagons.[10] Alternately one of the hexagonal tilings can remain regular, and the second one stretched and flattened by in each direction, intersecting into 2 forms of pentagons.

Cairo pentagonal tiling 2-colors.png Cairo pentagonal tiling 2-colors-concave.png Cairo tiling distorted regular hexagon.png

Topologically equivalent tilings[edit]

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.

Wallpaper group-p4g-1.jpg Wallpaper group-p4g-with Cairo pentagonal tiling2.png Wallpaper group-p4g-with Cairo pentagonal tiling.png
Basketweave tiling Cairo overlay

Truncated cairo pentagonal tiling[edit]

Truncating the 4-valence nodes creates a form related to the Goldberg polyhedra, and can be given the symbol {4+,4}2,1. The pentagons are truncated into heptagons; the dual {4,4+}2,1 has all triangle faces, related to the geodesic polyhedra. It can be seen as a snub square tiling with its squares replaced by 4 triangles.

Whirl square tiling.svg
Truncated cairo pentagonal tiling
Dual whirl square tiling.svg
Kis snub square tiling

Related polyhedra and tilings[edit]

The Cairo pentagonal tiling is similar to the prismatic pentagonal tiling with face configuration V3., and two 2-uniform dual tilings and 2 3-uniform duals which mix the two types of pentagons. They are drawn here with colored edges, or k-isohedral pentagons.[11]

33344 tiling face purple.png
33434 tiling face green.png
Related pentagonal tilings
Cairo pentagonal tiling 2-uniform duals
p4g (4*2) p2, (2222) pgg (22×) cmm (2*22)
1-uniform 9 dual edgecolor.svg 1 uniform 9 dual color1.png 2-uniform 17 dual edgecolor.svg 2-uniform 17 dual color2.png 2-uniform 16 dual edgecolor.svg 2-uniform 16 dual color2.png
V3. (V3.; V3.
Prismatic pentagonal tiling 3-uniform duals
cmm (2*22) p2 (2222) pgg (22×) p2 (2222) pgg (22×)
1-uniform 8 dual edgecolor.svg 1-uniform 8 dual color1.png 3-uniform 53 dual edgecolor.svg 3-uniform 53 dual color3.png 3-uniform 55 dual edgecolor.svg 3-uniform 55 dual color3.png
V3. (V3.; V3.

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

It is in a sequence of dual snub polyhedra and tilings with face configuration V3.3.n.3.n.

See also[edit]


  1. ^ 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.
  2. ^ Martin, George Edward (1982), Transformation Geometry: An Introduction to Symmetry, Undergraduate Texts in Mathematics, Springer, p. 119, ISBN 978-0-387-90636-2.
  3. ^ 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.
  4. ^ Macmahon, Major P. A. (1921), New Mathematical Pastimes, University Press. PDF [1] p.101
  5. ^ John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 [2] Archived 2010-09-19 at the Wayback Machine (Chapter 21, Naming Archimedean and Catalan polyhedra and tilings, p288 table)
  6. ^ Kotani, M.; Sunada, T. (2000), "Standard realizations of crystal lattices via harmonic maps", Transactions of the American Mathematical Society, 353: 1–20, doi:10.1090/S0002-9947-00-02632-5
  7. ^ T. Sunada, Topological Crystallography ---With a View Towards Discrete Geometric Analysis---, Surveys and Tutorials in the Applied Mathematical Sciences, Vol. 6, Springer
  8. ^
  9. ^ Weisstein, Eric W. "Dual tessellation". MathWorld.
  10. ^ Defining a cairo type tiling
  11. ^ Chavey, D. (1989). "Tilings by Regular Polygons—II: A Catalog of Tilings". Computers & Mathematics with Applications. 17: 147–165. doi:10.1016/0898-1221(89)90156-9.

Further reading[edit]

  • Grünbaum, Branko; 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.
  • Keith Critchlow, Order in Space: A design source book, 1970, p. 77-76, pattern 3

External links[edit]