Cairo pentagonal tiling

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Cairo pentagonal tiling
Equilateral Cairo tiling.svg
Equilateral form of the Cairo 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]

Contents

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]

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 L1 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 in which, after the first three terms, each term differs by 16 from the term three steps back in the sequence. One can also define analogous coordination sequences for the vertices of the tiling instead of for its tiles, but because there are two types of vertices (of degree three and degree four) there are two different coordination sequences arising in this way. The degree-four sequence is the same as for the square grid. [11] [12]

Special cases

Catalan tiling

P2 dual.png
Cairo tiling as the dual of the snub square tiling
Pentagonal Cairo Snub Square Tile 2.svg
Geometry of pentagons for the dual snub square 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

Collinear form of Cairo tiling, with integer-coordinate pentagons, formed by flattening two perpendicular regular hexagonal tilings in perpendicular directions Collinear Cairo tiling.svg
Collinear form of Cairo tiling, with integer-coordinate pentagons, formed by flattening two perpendicular regular hexagonal tilings in perpendicular directions

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]

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]

Related Research Articles

In geometry, an octahedron is a polyhedron with eight faces. An octahedron can be considered as a square bipyramid. When the edges of a square bipyramid are all equal in length, it produces a regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. It is also an example of a deltahedron. An octahedron is the three-dimensional case of the more general concept of a cross polytope.

<span class="mw-page-title-main">Hexagon</span> Shape with six sides

In geometry, a hexagon is a six-sided polygon. The total of the internal angles of any simple (non-self-intersecting) hexagon is 720°.

<span class="mw-page-title-main">Kite (geometry)</span> Quadrilateral symmetric across a diagonal

In Euclidean geometry, a kite is a quadrilateral with reflection symmetry across a diagonal. Because of this symmetry, a kite has two equal angles and two pairs of adjacent equal-length sides. Kites are also known as deltoids, but the word deltoid may also refer to a deltoid curve, an unrelated geometric object sometimes studied in connection with quadrilaterals. A kite may also be called a dart, particularly if it is not convex.

<span class="mw-page-title-main">Truncated icosidodecahedron</span> Archimedean solid

In geometry, a truncated icosidodecahedron, rhombitruncated icosidodecahedron, great rhombicosidodecahedron, omnitruncated dodecahedron or omnitruncated icosahedron is an Archimedean solid, one of thirteen convex, isogonal, non-prismatic solids constructed by two or more types of regular polygon faces.

<span class="mw-page-title-main">Tessellation</span> Tiling of a plane in mathematics

A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called tiles, with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of geometries.

<span class="mw-page-title-main">Euclidean tilings by convex regular polygons</span> Subdivision of the plane into polygons that are all regular

Euclidean plane tilings by convex regular polygons have been widely used since antiquity. The first systematic mathematical treatment was that of Kepler in his Harmonices Mundi.

<span class="mw-page-title-main">Hexagonal bipyramid</span> Polyhedron; 2 hexagonal pyramids joined base-to-base

A hexagonal bipyramid is a polyhedron formed from two hexagonal pyramids joined at their bases. The resulting solid has 12 triangular faces, 8 vertices and 18 edges. The 12 faces are identical isosceles triangles.

<span class="mw-page-title-main">Hexagonal tiling</span> Regular tiling of a two-dimensional space

In geometry, the hexagonal tiling or hexagonal tessellation is a regular tiling of the Euclidean plane, in which exactly three hexagons meet at each vertex. It has Schläfli symbol of {6,3} or t{3,6} .

<span class="mw-page-title-main">Truncated hexagonal tiling</span>

In geometry, the truncated hexagonal tiling is a semiregular tiling of the Euclidean plane. There are 2 dodecagons (12-sides) and one triangle on each vertex.

<span class="mw-page-title-main">Rhombitrihexagonal tiling</span> Semiregular tiling of the Euclidean plane

In geometry, the rhombitrihexagonal tiling is a semiregular tiling of the Euclidean plane. There are one triangle, two squares, and one hexagon on each vertex. It has Schläfli symbol of rr{3,6}.

<span class="mw-page-title-main">Snub trihexagonal tiling</span>

In geometry, the snub hexagonal tiling is a semiregular tiling of the Euclidean plane. There are four triangles and one hexagon on each vertex. It has Schläfli symbol sr{3,6}. The snub tetrahexagonal tiling is a related hyperbolic tiling with Schläfli symbol sr{4,6}.

<span class="mw-page-title-main">Elongated triangular tiling</span>

In geometry, the elongated triangular tiling is a semiregular tiling of the Euclidean plane. There are three triangles and two squares on each vertex. It is named as a triangular tiling elongated by rows of squares, and given Schläfli symbol {3,6}:e.

<span class="mw-page-title-main">Pentagonal tiling</span> A tiling of the plane by pentagons

In geometry, a pentagonal tiling is a tiling of the plane where each individual piece is in the shape of a pentagon.

In geometry, a polytope or a tiling is isotoxal or edge-transitive if its symmetries act transitively on its edges. Informally, this means that there is only one type of edge to the object: given two edges, there is a translation, rotation, and/or reflection that will move one edge to the other while leaving the region occupied by the object unchanged.

<span class="mw-page-title-main">Laves graph</span> Periodic spatial graph

In geometry and crystallography, the Laves graph is an infinite and highly symmetric system of points and line segments in three-dimensional Euclidean space, forming a periodic graph. Three equal-length segments meet at 120° angles at each point, and all cycles use ten or more segments. It is the shortest possible triply periodic graph, relative to the volume of its fundamental domain. One arrangement of the Laves graph uses one out of every eight of the points in the integer lattice as its points, and connects all pairs of these points that are nearest neighbors, at distance . It can also be defined, divorced from its geometry, as an abstract undirected graph, a covering graph of the complete graph on four vertices.

<span class="mw-page-title-main">Planigon</span> Convex polygon which can tile the plane by itself

In geometry, a planigon is a convex polygon that can fill the plane with only copies of itself. In the Euclidean plane there are 3 regular planigons; equilateral triangle, squares, and regular hexagons; and 8 semiregular planigons; and 4 demiregular planigons which can tile the plane only with other planigons.

In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,7,3}.

In the geometry of hyperbolic 3-space, the order-8-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,8,3}.

In the geometry of hyperbolic 3-space, the order-infinite-3 triangular honeycomb is a regular space-filling tessellation with Schläfli symbol {3,∞,3}.

References

  1. 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 .
  2. Macmahon, Major P. A. (1921), New Mathematical Pastimes, University Press, p. 101
  3. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008), The Symmetries of Things, AK Peters, p.  288, ISBN   978-1-56881-220-5
  4. 1 2 3 4 5 6 7 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
  5. Rao, Michaël (2017), Exhaustive search of convex pentagons which tile the plane (PDF), arXiv: 1708.00274
  6. 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
  7. 1 2 Schattschneider, Doris; Walker, Wallace (1977), "Dodecahedron", M. C. Escher Kaleidocycles, Ballantine Books, p. 22; reprinted by Taschen, 2015
  8. 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
  9. 1 2 3 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, S2CID   125608794
  10. 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
  11. 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
  12. 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, archived from the original (PDF) on 2022-02-17, retrieved 2021-06-18
  13. 1 2 Rollett, A. P. (September 1955), "2530. A pentagonal tessellation", Mathematical Notes, The Mathematical Gazette , 39 (329): 209, doi:10.2307/3608750, JSTOR   3608750, S2CID   250439435
  14. 1 2 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
  15. 1 2 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
  16. 1 2 3 4 Bailey, David, "Cairo tiling", David Bailey's World of Escher-like Tessellations, archived from the original on 2020-12-03, retrieved 2020-12-06
  17. Dunn, J. A. (December 1971), "Tessellations with pentagons", The Mathematical Gazette , 55 (394): 366–369, doi:10.2307/3612359, JSTOR   3612359, S2CID   118680100 . Although Dunn writes that the equilateral form of the tiling was used in Cairo, this appears to be a mistake.
  18. 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 .
  19. Martin, George Edward (1982), Transformation Geometry: An Introduction to Symmetry, Undergraduate Texts in Mathematics, Springer, p. 119, ISBN   978-0-387-90636-2 .
  20. 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, S2CID   198468426
  21. Nisbet, Harry (1906), Grammar of Textile Design, London: Scott, Greenwood & Son, p. 101
  22. Moore, H. C. (July 20, 1909), Tile (US Patent 928,320)
  23. 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.
  24. 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
  25. 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
  26. 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