Cairo pentagonal tiling

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Cairo pentagonal tiling
1-uniform 9 dual.svg
Type Dual semiregular tiling
Coxeter diagram CDel 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
Faces irregular pentagons
Face configuration V3.
Tiling face 3-3-4-3-4.svg
Symmetry group p4g, [4+,4], (4*2)
p4, [4,4]+, (442)
Rotation group p4, [4,4]+, (442)
Dual Snub square tiling
Properties face-transitive

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 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]

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  \sqrt 3. 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  \sqrt 3 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

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
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.

4n2 symmetry mutations of snub tilings:
Spherical Euclidean Compact hyperbolic Paracomp.
242 342 442 542 642 742 842 ∞42
Spherical square antiprism.png Spherical snub cube.png Uniform tiling 44-snub.png Uniform tiling 54-snub.png Uniform tiling 64-snub.png Uniform tiling 74-snub.png Uniform tiling 84-snub.png Uniform tiling i42-snub.png
Spherical tetragonal trapezohedron.png Spherical pentagonal icositetrahedron.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg Order-5-4 floret pentagonal tiling.png
Config. V3. V3. V3. V3. V3. V3. V3. V3.3.4.3.∞

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

4n2 symmetry mutations of snub tilings: 3.3.n.3.n
Spherical Euclidean Compact hyperbolic Paracompact
222 322 442 552 662 772 882 ∞∞2
Digonal antiprism.png Pseudoicosahedron-3.png Uniform tiling 44-snub.png Uniform tiling 552-snub.png Uniform tiling 66-snub.png Uniform tiling 77-snub.png Uniform tiling 88-snub.png Uniform tiling ii2-snub.png
Config. 3.3.∞.3.∞
Digonal trapezohedron.png Pyritohedron.png Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg Infinitely-infinite-order floret pentagonal tiling.png
Config. V3. V3. V3. V3. V3. V3. V3. V3.3.∞.3.∞

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] (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", Transaction of American Mathematical Society 353: 1–20 
  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. 

Additional reading[edit]

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

External links[edit]