Cairo pentagonal tiling
| Cairo pentagonal tiling | |
|---|---|
 |
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| Type | Dual semiregular tiling |
| Coxeter-Dynkin diagram |    |
| 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, and their angles in sequence are 120°, 120°, 90°, 120°, 90°.
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Dual tiling [edit]
It is the dual of the snub square tiling, made of two squares and three equilateral triangles around each vertex.[6]
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.
 Basketweave tiling |
 Cairo tiling overlay |
 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.
| Symmetry 4n2 [n,4]+ |
Spherical | Euclidean | Hyperbolic | |||||
|---|---|---|---|---|---|---|---|---|
| 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 |
 3.3.4.3.2 |
 3.3.4.3.3 |
 3.3.4.3.4 |
 3.3.4.3.5 |
 3.3.4.3.6 |
 3.3.4.3.7 |
 3.3.4.3.8 |
 3.3.4.3.∞ |
| Coxeter Schläfli |
  s{2,4} |
 s{3,4} |
s{4,4} |
 s{5,4} |
 s{6,4} |
 s{7,4} |
 s{8,4} |
 s{∞,4} |
| Snub dual figure |
 V3.3.4.3.2 |
 V3.3.4.3.3 |
V3.3.4.3.4 |
 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 | |
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See also [edit]
Notes [edit]
- ^ 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, 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)
- ^ Weisstein, Eric W., "Dual tessellation", MathWorld.
References [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]
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