# Cairo pentagonal tiling Information

*https://en.wikipedia.org/wiki/Cairo_pentagonal_tiling*

Cairo pentagonal tiling | |
---|---|

Type | Pentagonal tiling |

Faces | irregular pentagons |

Dual polyhedron | Snub square tiling |

Properties | face-transitive |

In
geometry, a **Cairo pentagonal tiling** is a
tessellation of the
Euclidean plane by congruent convex
pentagons, formed by overlaying two tessellations of the plane by hexagons and named for its use as a paving design in
Cairo. It is also called **MacMahon's net**^{
[1]} after
Percy Alexander MacMahon, who depicted it in his 1921 publication *New Mathematical Pastimes*.^{
[2]}
John Horton Conway called it a **4-fold pentille**.^{
[3]}

Infinitely many different pentagons can form this pattern, belonging to two of the 15 families of convex pentagons that can tile the plane. Their tilings have varying symmetries; all are face-symmetric. One particular form of the tiling, dual to the snub square tiling, has tiles with the minimum possible perimeter among all pentagonal tilings. Another, overlaying two flattened tilings by regular hexagons, is the form used in Cairo and has the property that every edge is collinear with infinitely many other edges.

In architecture, beyond Cairo, the Cairo tiling has been used in Mughal architecture in 18th-century India, in the early 20th-century Laeiszhalle in Germany, and in many modern buildings and installations. It has also been studied as a crystal structure and appears in the art of M. C. Escher.

## Structure and classification

The union of all edges of a Cairo tiling is the same as the union of two tilings of the plane by
hexagons. Each hexagon of one tiling surrounds two vertices of the other tiling, and is divided by the hexagons of the other tiling into four of the pentagons in the Cairo tiling.^{
[4]} Infinitely many different pentagons can form Cairo tilings, all with the same pattern of adjacencies between tiles and with the same decomposition into hexagons, but with varying edge lengths, angles, and symmetries. The pentagons that form these tilings can be grouped into two different infinite families, drawn from the 15 families of
convex pentagons that can tile the plane,^{
[5]} and the five families of pentagon found by
Karl Reinhardt in 1918 that can tile the plane
isohedrally (all tiles symmetric to each other).^{
[6]}

One of these two families consists of pentagons that have two non-adjacent
right angles, with a pair of sides of equal length meeting at each of these right angles. Any pentagon meeting these requirements tiles the plane by copies that, at the chosen right angled corners, are rotated by a right angle with respect to each other. At the pentagon sides that are not adjacent to one of these two right angles, two tiles meet, rotated by a 180° angle with respect to each other. The result is an isohedral tiling, meaning that any pentagon in the tiling can be transformed into any other pentagon by a symmetry of the tiling. These pentagons and their tiling are often listed as "type 4" in the listing of types of pentagon that can tile.^{
[4]} For any type 4 Cairo tiling, twelve of the same tiles can also cover the surface of a cube, with one tile folded across each cube edge and three right angles of tiles meeting at each cube vertex, to form the same combinatorial structure as a regular
dodecahedron.^{
[7]}^{
[8]}

The other family of pentagons forming the Cairo tiling are pentagons that have two
complementary angles at non-adjacent vertices, each having the same two side lengths incident to it. In their tilings, the vertices with complementary angles alternate around each degree-four vertex. The pentagons meeting these constraints
are not generally listed as one of the 15 families of pentagons that tile; rather, they are part of a larger family of pentagons (the "type 2" pentagons) that tile the plane isohedrally in a different way.^{
[4]}

Bilaterally symmetric Cairo tilings are formed by pentagons that belong to both the type 2 and type 4 families.^{
[4]} The basketweave
brick paving pattern can be seen as a
degenerate case of the bilaterally symmetric Cairo tilings, with each brick (a rectangle) interpreted as a pentagon with four right angles and one 180° angle.^{
[9]}

Type 2 Cairo tiles have non-adjacent complementary angles, with the same two adjacent side lengths

Type 4 tiles have non-adjacent right angles between pairs of equal-length sides

In a type 4 chiral tiling, the pentagons can be bilaterally symmetric even when the tiling isn't

It is possible to assign six-dimensional
half-integer coordinates to the pentagons of the tiling, in such a way that the number of edge-to-edge steps between any two pentagons equals the
*L*^{1} distance between their coordinates. The six coordinates of each pentagon can be grouped into two triples of coordinates, in which each triple gives the coordinates of a hexagon in an analogous three-dimensional coordinate system for each of the two overlaid hexagon tilings.^{
[10]} The number of tiles that are steps away from any given tile, for , is given by the
coordination sequence

^{ [11]}

^{ [12]}

## Special cases

### Catalan tiling

The
snub square tiling, made of two squares and three equilateral triangles around each vertex, has a bilaterally symmetric Cairo tiling as its
dual tiling.^{
[13]} The Cairo tiling can be formed from the snub square tiling by placing a vertex of the Cairo tiling at the center of each square or triangle of the snub square tiling, and connecting these vertices by edges when they come from adjacent tiles.^{
[14]} Its pentagons can be
circumscribed around a circle. They have four long edges and one short one with lengths in the ratio . The angles of these pentagons form the sequence 120°, 120°, 90°, 120°, 90°.^{
[15]}

The snub square tiling is an
Archimedean tiling, and as the dual to an Archimedean tiling this form of the Cairo pentagonal tiling is a
Catalan tiling or Laves tiling.^{
[14]} It is one of two monohedral pentagonal tilings that, when the tiles have unit area, minimizes the perimeter of the tiles. The other is also a tiling by circumscribed pentagons with two right angles and three 120° angles, but with the two right angles adjacent; there are also infinitely many tilings formed by combining both kinds of pentagon.^{
[15]}

### Tilings with collinear edges

Pentagons with integer vertex coordinates , , and , with four equal sides shorter than the remaining side, form a Cairo tiling whose two hexagonal tilings can be formed by flattening two perpendicular
tilings by regular hexagons in perpendicular directions, by a ratio of . This form of the Cairo tiling inherits the property of the tilings by regular hexagons (unchanged by the flattening), that every edge is collinear with infinitely many other edges.^{
[9]}^{
[16]}

### Tilings with equal side lengths

The regular pentagon cannot form Cairo tilings, as it does not tile the plane without gaps. There is a unique
equilateral pentagon that can form a type 4 Cairo tiling; it has five equal sides but its angles are unequal, and its tiling is bilaterally symmetric.^{
[4]}^{
[13]} Infinitely many other equilateral pentagons can form type 2 Cairo tilings.^{
[4]}

## Applications

Several streets in
Cairo have been paved with the collinear form of the Cairo tiling;^{
[9]}^{
[17]} this application is the origin of the name of the tiling.^{
[18]}^{
[19]} As of 2019 this pattern can still be seen as a surface decoration for square tiles near the
Qasr El Nil Bridge and the
El Behoos Metro station; other versions of the tiling are visible elsewhere in the city.^{
[20]} Some authors including
Martin Gardner have written that this pattern is used more widely in
Islamic architecture, and although this claim appears to have been based on a misunderstanding, patterns resembling the Cairo tiling are visible on the 17th-century
Tomb of I'timād-ud-Daulah in India, and the Cairo tiling itself has been found on a 17th-century
Mughal
jali.^{
[16]}

Tomb of I'timād-ud-Daulah, with rectangular side panels resembling the Cairo tiling

Centar Zamet, with the Cairo tiling visible on its walls

Cairo tiling in Hørsholm, Denmark

One of the earliest publications on the Cairo tiling as a decorative pattern occurs in a book on
textile design from 1906.^{
[21]} Inventor H. C. Moore filed a US patent on tiles forming this pattern in 1908.^{
[22]} At roughly the same time,
Villeroy & Boch created a line of ceramic floor tiles in the Cairo tiling pattern, used in the foyer of the
Laeiszhalle in
Hamburg, Germany. The Cairo tiling has been used as a decorative pattern in many recent architectural designs; for instance, the city center of
Hørsholm, Denmark, is paved with this pattern, and the
Centar Zamet, a sports hall in Croatia, uses it both for its exterior walls and its paving tiles.^{
[16]}

In
crystallography, this tiling has been studied at least since 1911.^{
[23]} It has been proposed as the structure for layered
hydrate crystals,^{
[24]} certain compounds of
bismuth and
iron,^{
[25]} and
penta-graphene, a hypothetical
compound of pure carbon. In the penta-graphene structure, the edges of the tiling incident to degree-four vertices form
single bonds, while the remaining edges form
double bonds. In its
hydrogenated form, penta-graphane, all bonds are single bonds and the carbon atoms at the degree-three vertices of the structure have a fourth bond connecting them to hydrogen atoms.^{
[26]}

The Cairo tiling has been described as one of
M. C. Escher's "favorite geometric patterns".^{
[7]} He used it as the basis for his drawing *Shells and Starfish* (1941), in the bees-on-flowers segment of his *
Metamorphosis III* (1967–1968), and in several other drawings from 1967–1968. An image of this tessellation has also been used as the cover art for the 1974 first edition of
H. S. M. Coxeter's book *Regular Complex Polytopes*.^{
[4]}^{
[16]}

## References

**^**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, Bibcode: 1980RSPTA.295..553O, doi: 10.1098/rsta.1980.0150, JSTOR 36648, S2CID 121456259.**^**Macmahon, Major P. A. (1921),*New Mathematical Pastimes*, University Press, p. 101**^**Conway, John H.; Burgiel, Heidi; Goodman-Strass, Chaim (2008),*The Symmetries of Things*, AK Peters, p. 288, ISBN 978-1-56881-220-5- ^
^{a}^{b}^{c}^{d}^{e}^{f}^{g}Schattschneider, Doris (1978), "Tiling the plane with congruent pentagons",*Mathematics Magazine*,**51**(1): 29–44, doi: 10.1080/0025570X.1978.11976672, JSTOR 2689644, MR 0493766 **^**Rao, Michaël (2017),*Exhaustive search of convex pentagons which tile the plane*(PDF), arXiv: 1708.00274**^**Reinhardt, Karl (1918),*Über die Zerlegung der Ebene in Polygone*(Doctoral dissertation) (in German), Borna-Leipzig: Druck von Robert Noske, "Vierter Typus", p. 78, and Figure 24, p. 81- ^
^{a}^{b}Schattschneider, Doris; Walker, Wallace (1977), "Dodecahedron",*M. C. Escher Kaleidocycles*, Ballantine Books, p. 22; reprinted by Taschen, 2015 **^**Thomas, B.G.; Hann, M.A. (2008), "Patterning by projection: Tiling the dodecahedron and other solids", in Sarhangi, Reza; Séquin, Carlo H. (eds.),*Bridges Leeuwarden: Mathematics, Music, Art, Architecture, Culture*, London: Tarquin Publications, pp. 101–108, ISBN 9780966520194- ^
^{a}^{b}^{c}Macmillan, R. H. (December 1979), "Pyramids and pavements: Some thoughts from Cairo",*The Mathematical Gazette*,**63**(426): 251–255, doi: 10.2307/3618038, JSTOR 3618038 **^**Kovács, Gergely; Nagy, Benedek; Turgay, Neşet Deniz (May 2021), "Distance on the Cairo pattern",*Pattern Recognition Letters*,**145**: 141–146, Bibcode: 2021PaReL.145..141K, doi: 10.1016/j.patrec.2021.02.002, S2CID 233375125**^**Coordination sequences for the Cairo pentagonal tiling in the On-Line Encyclopedia of Integer Sequences: A219529 for pentagons, A296368 for degree-three vertices, and A008574 for degree-four vertices, retrieved 2021-06-17**^**Goodman-Strauss, C.; Sloane, N. J. A. (2019), "A coloring-book approach to finding coordination sequences" (PDF),*Acta Crystallographica Section A*,**75**(1): 121–134, arXiv: 1803.08530, doi: 10.1107/s2053273318014481, MR 3896412, PMID 30575590, S2CID 4553572- ^
^{a}^{b}Rollett, A. P. (September 1955), "2530. A pentagonal tessellation", Mathematical Notes,*The Mathematical Gazette*,**39**(329): 209, doi: 10.2307/3608750, JSTOR 3608750 - ^
^{a}^{b}Steurer, Walter; Dshemuchadse, Julia (2016),*Intermetallics: Structures, Properties, and Statistics*, International Union of Crystallography Monographs on Crystallography, vol. 26, Oxford University Press, p. 42, ISBN 9780191023927 - ^
^{a}^{b}Chung, Ping Ngai; Fernandez, Miguel A.; Li, Yifei; Mara, Michael; Morgan, Frank; Plata, Isamar Rosa; Shah, Niralee; Vieira, Luis Sordo; Wikner, Elena (2012), "Isoperimetric pentagonal tilings",*Notices of the American Mathematical Society*,**59**(5): 632–640, doi: 10.1090/noti838, MR 2954290 - ^
^{a}^{b}^{c}^{d}Bailey, David, "Cairo tiling",*David Bailey's World of Escher-like Tessellations*, retrieved 2020-12-06 **^**Dunn, J. A. (December 1971), "Tessellations with pentagons",*The Mathematical Gazette*,**55**(394): 366–369, doi: 10.2307/3612359, JSTOR 3612359. Although Dunn writes that the equilateral form of the tiling was used in Cairo, this appears to be a mistake.**^**Alsina, Claudi; Nelsen, Roger B. (2010),*Charming proofs: a journey into elegant mathematics*, Dolciani mathematical expositions, vol. 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.**^**Morgan, Frank (2019), "My undercover mission to find Cairo tilings",*The Mathematical Intelligencer*,**41**(3): 19–22, doi: 10.1007/s00283-019-09906-7, MR 3995312**^**Nisbet, Harry (1906),*Grammar of Textile Design*, London: Scott, Greenwood & Son, p. 101**^**Moore, H. C. (July 20, 1909),*Tile*(US Patent 928,320)**^**Haag, F. (1911), "Die regelmäßigen Planteilungen",*Zeitschrift für Kristallographie, Kristallgeometrie, Kristallphysik, Kristallchemie*,**49**: 360–369, hdl: 2027/uc1.b3327994 See in particular Figures 2b, p. 361, and 4a, p. 363.**^**Banaru, A. M.; Banaru, G. A. (August 2011), "Cairo tiling and the topology of layered hydrates",*Moscow University Chemistry Bulletin*,**66**(3), Article 159, doi: 10.3103/S0027131411030023, S2CID 96002269**^**Ressouche, E.; Simonet, V.; Canals, B.; Gospodinov, M.; Skumryev, V. (December 2009), "Magnetic frustration in an iron-based Cairo pentagonal lattice",*Physical Review Letters*,**103**(26): 267204, arXiv: 1001.0710, Bibcode: 2009PhRvL.103z7204R, doi: 10.1103/physrevlett.103.267204, PMID 20366341, S2CID 20752605**^**Zhang, Shunhong; Zhou, Jian; Wang, Qian; Chen, Xiaoshuang; Kawazoe, Yoshiyuki; Jena, Puru (February 2015), "Penta-graphene: A new carbon allotrope",*Proceedings of the National Academy of Sciences of the United States of America*,**112**(8): 2372–2377, Bibcode: 2015PNAS..112.2372Z, doi: 10.1073/pnas.1416591112, PMC 4345574, PMID 25646451