{"id":2204,"date":"2016-01-15T01:00:08","date_gmt":"2016-01-15T01:00:08","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=2204"},"modified":"2016-01-23T18:42:39","modified_gmt":"2016-01-23T18:42:39","slug":"cairo-tiling","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2016\/01\/15\/cairo-tiling\/","title":{"rendered":"Cairo Tiling"},"content":{"rendered":"
\n
\"Cairo<\/a>

Cairo Tiling – Tom Ruen<\/p><\/div>\n<\/div>\n

The Cairo tiling<\/a> is a tiling of the plane by pentagons. <\/p>\n

To construct the Cairo tiling, we can start with a snub square tiling<\/a>:<\/p>\n

\"Snub<\/a>

Snub Square Tiling – Tom Ruen<\/p><\/div>\n

The snub square tiling is a uniform tiling<\/a> of the plane by squares and equilateral triangles, with 2 squares and 3 triangles meeting at each vertex, arranged in the pattern \\(3.3.4.3.4\\). <\/p>\n

Starting from the snub square tiling, form the dual tiling<\/a>. This has a vertex at the center of each square or triangle, with one edge crossing each edge of the snub square tiling:<\/p>\n

\"Snub<\/a>

Snub Square Tiling and Dual – TED-43<\/p><\/div>\n

The result is the Cairo tiling!<\/p>\n

This tiling gets its name because it is apparently used to tile some streets in Cairo.<\/p>\n

Puzzle 1.<\/b> Is this really true? Which streets? Has someone taken a photograph?<\/p>\n

Puzzle 2.<\/b> What are the internal angles and side lengths of the pentagons in the Cairo tiling?<\/p>\n

Carbon can form flat molecular sheets consisting of regular hexagons, called graphene<\/a>. In February 2015, this article argued that carbon could also form sheets in which the atoms lay at the vertices of a Cairo tiling:<\/p>\n

• 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<\/a><\/i> 112<\/b> (2015), 2372–2377.<\/p>\n

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:<\/p>\n

• 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<\/a>, Proceedings of the National Academy of Sciences<\/i> 112<\/b> (2015), 15609\u201315612.<\/p>\n

This paper has beautiful pictures showing how penta-graphene could transform into ordinary graphene. You can read a summary of the dispute here:<\/p>\n

• Heather Zeiger, Criteria to predict experimentally stable allotropes<\/a>, Phys.org<\/i>, 5 January 2016.<\/p>\n

The above picture of the Cairo tiling was made by Tom Ruen<\/a> and placed it on Wikicommons<\/a> with a Creative Commons Attribution-Share Alike 4.0 International<\/a> license. Tom Ruen also created the picture of the snub square tiling and placed it on Wikicommons<\/a> 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<\/a>, and he placed it on Wikicommons<\/a> with a GNU Free Documentation License, version 1.2<\/a>.<\/p>\n

\n

Visual Insight<\/i> is a place to share striking images that help explain advanced topics in mathematics. I\u2019m always looking for truly beautiful images, so if you know about one, please drop a comment here<\/a> and let me know!<\/p>\n

<\/div>","protected":false},"excerpt":{"rendered":"

The Cairo tiling<\/a> is a tiling of the plane by non-regular pentagons which is dual to the snub square tiling<\/a>.<\/p>\n

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