Cairo pentagonal tiling

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Cairo pentagonal tiling
Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.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 V3.3.4.3.4
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 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]

This tiling can be seen as the union of two flattened perpendicular hexagonal tilings. Each hexagon is divided into four pentagons. 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°.

Dual tiling[edit]

It is the dual of the snub square tiling, made of two squares and three equilateral triangles around each vertex.[7]

P2 dual.png

Related polyhedra and 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
Basketweave tiling
Wallpaper group-p4g-with Cairo pentagonal tiling.png
Cairo tiling overlay
Cornish Slate Floor, Twill pattern.jpg
Slate floor

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

Dimensional family of snub polyhedra and tilings: 3.3.4.3.n
Symmetry
4n2
[n,4]+
Spherical Euclidean Compact hyperbolic Paracompact
242
[2,4]+
342
[3,4]+
442
[4,4]+
542
[5,4]+
642
[6,4]+
742
[7,4]+
842
[8,4]+...
∞42
[∞,4]+
Snub
figure
Spherical square antiprism.png
3.3.4.3.2
Spherical snub cube.png
3.3.4.3.3
Uniform tiling 44-snub.png
3.3.4.3.4
Uniform tiling 54-snub.png
3.3.4.3.5
Uniform tiling 64-snub.png
3.3.4.3.6
Uniform tiling 74-snub.png
3.3.4.3.7
Uniform tiling 84-snub.png
3.3.4.3.8
Uniform tiling i42-snub.png
3.3.4.3.∞
Coxeter
Schläfli
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{2,4}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{3,4}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{4,4}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{5,4}
CDel node h.pngCDel 6.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{6,4}
CDel node h.pngCDel 7.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{7,4}
CDel node h.pngCDel 8.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{8,4}
CDel node h.pngCDel infin.pngCDel node h.pngCDel 4.pngCDel node h.png
sr{∞,4}
Snub
dual
figure
Tetragonal trapezohedron.png
V3.3.4.3.2
Pentagonalicositetrahedronccw.jpg
V3.3.4.3.3
Tiling Dual Semiregular V3-3-4-3-4 Cairo Pentagonal.svg
V3.3.4.3.4
Order-5-4 floret pentagonal tiling.png
V3.3.4.3.5
V3.3.4.3.6 V3.3.4.3.7 V3.3.4.3.8 V3.3.4.3.∞
Coxeter CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 4.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 4.pngCDel node fh.png

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 .
  5. ^ 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)
  6. ^ http://catnaps.org/islamic/geometry2.html
  7. ^ Weisstein, Eric W., "Dual tessellation", MathWorld.

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.

External links[edit]