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 configurationV3.3.4.3.4
Tiling face 3-3-4-3-4.svg
Propertiesface-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 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[edit]

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 V3.3.4.3.4.

It is similar to the prismatic pentagonal tiling with face configuration V3.3.3.4.4, which has its right angles adjacent to each other.

Variations[edit]

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
B=D=90°
Prototile p5-type8.png
b=c=d=e
2B+C=D+2E=360°
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 V3.3.3.4.4, 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
V3.3.3.4.4
33434 tiling face green.png
V3.3.4.3.4
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
V3.3.4.3.4 (V3.3.3.4.4; V3.3.4.3.4)
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
V3.3.3.4.4 (V3.3.3.4.4; V3.3.4.3.4)

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]

Notes[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. ^ http://catnaps.org/islamic/geometry2.html
  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]