Cairo pentagonal tiling

Cairo pentagonal tiling
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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

, 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

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

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]

case of the bilaterally symmetric Cairo tilings, with each brick (a ${\displaystyle 1\times 2}$ 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
• 
  Bilaterally symmetric tilings (belonging to both types) use tiles with non-adjacent right angles and four equal edges

• 
  Type 2 Cairo tiling, with coloring showing reflected and non-reflected tiles
• 
  In a type 4 Cairo tiling, the pentagons can be bilaterally symmetric even when the tiling isn't
• 
  The basketweave, a degenerate bilaterally symmetric tiling, with non-degenerate tiling overlaid

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 ${\displaystyle i}$ steps away from any given tile, for ${\displaystyle i=0,1,2,\dots }$, is given by the coordination sequence $${\displaystyle 1,5,11,16,21,27,32,37,\dots }$$ 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

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 ${\displaystyle 1:{\sqrt {3}}-1}$. The angles of these pentagons form the sequence 120°, 120°, 90°, 120°, 90°.^[15]

The snub square tiling is an

Tilings with collinear edges

Pentagons with integer vertex coordinates ${\displaystyle (\pm 2,0)}$, ${\displaystyle (\pm 3,3)}$, and ${\displaystyle (0,4)}$, 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 ${\displaystyle {\sqrt {3}}}$. 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

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