The Cairo tiling is a tiling of the plane by pentagons.
To construct the Cairo tiling, we can start with a snub square tiling:
The snub square tiling is a uniform tiling of the plane by squares and equilateral triangles, with 2 squares and 3 triangles meeting at each vertex, arranged in the pattern \(126.96.36.199.4\).
Starting from the snub square tiling, form the dual tiling. This has a vertex at the center of each square or triangle, with one edge crossing each edge of the snub square tiling:
The result is the Cairo tiling!
This tiling gets its name because it is apparently used to tile some streets in Cairo.
Puzzle 1. Is this really true? Which streets? Has someone taken a photograph?
Puzzle 2. What are the internal angles and side lengths of the pentagons in the Cairo tiling?
Carbon can form flat molecular sheets consisting of regular hexagons, called graphene. In February 2015, this article argued that carbon could also form sheets in which the atoms lay at the vertices of a Cairo tiling:
• Shunhong Zhang, Jian Zhou, Qian Wanga, Xiaoshuang Chen, Yoshiyuki Kawazoe and Puru Jena, Penta-graphene: a new carbon allotrope, Proceedings of the National Academy of Sciences 112 (2015), 2372–2377.
The authors did calculations to show that this hypothetical material would be stable. However, more recently a paper has come out arguing for the opposite conclusion:
• Christopher P. Ewels, Xavier Rocquefelte, Harold W. Kroto, Mark J. Rayson, Patrick R. Briddon, and Malcolm I. Heggie, Predicting experimentally stable allotropes: instability of penta-graphene, Proceedings of the National Academy of Sciences 112 (2015), 15609–15612.
This paper has beautiful pictures showing how penta-graphene could transform into ordinary graphene. You can read a summary of the dispute here:
• Heather Zeiger, Criteria to predict experimentally stable allotropes, Phys.org, 5 January 2016.
The above picture of the Cairo tiling was made by Tom Ruen and placed it on Wikicommons with a Creative Commons Attribution-Share Alike 4.0 International license. Tom Ruen also created the picture of the snub square tiling and placed it on Wikicommons with the same license. The picture of the square snub tiling together with its dual was made by a German Wikicommons user going by the name of TED-43, and he placed it on Wikicommons with a GNU Free Documentation License, version 1.2.
Visual Insight is a place to share striking images that help explain advanced topics in mathematics. I’m always looking for truly beautiful images, so if you know about one, please drop a comment here and let me know!