{"id":946,"date":"2014-07-15T01:00:22","date_gmt":"2014-07-15T01:00:22","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=946"},"modified":"2018-10-12T21:08:27","modified_gmt":"2018-10-12T21:08:27","slug":"73-tiling","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2014\/07\/15\/73-tiling\/","title":{"rendered":"{7,3} Tiling"},"content":{"rendered":"
\n
\"{7,3}<\/a>

{7,3} Tiling – Anton Sherwood<\/p><\/div>\n<\/div>\n

This is the {7,3} tiling<\/a><\/b>: a tiling of the hyperbolic plane by equal-sized regular heptagons, 3 meeting at each vertex. The symmetry group of this tiling is the Coxeter group<\/p>\n

\no—7—o—3—o<\/b>\n<\/div>\n

 <\/p>\n

which is generated by 3 reflections of the hyperbolic plane $s_1, s_2, s_3,$ obeying relations encoded in the edges of the diagram:<\/p>\n

$$ (s_1 s_2)^7 = 1 $$
\n$$ (s_2 s_3)^3 = 1 $$<\/p>\n

together with relations saying that each generator squares to 1 and distant ones commute:<\/p>\n

$$ s_1 s_3 = s_3 s_1 $$<\/p>\n

This group, also known as the (2,3,7) triangle group<\/b> or $\\Delta(2,3,7)$, is connected to a lot of interesting mathematics:<\/p>\n

• (2,3,7) triangle group<\/a>, Wikipedia.<\/p>\n

For example, Klein’s quartic curve, the maximally symmetric 3-holed Riemann surface, can be tiled by 24 regular heptagons. The best way to see this is to describe Klein’s quartic curve as a quotient of the hyperbolic plane by a discrete group of symmetries that preserves the {7,3,3} tiling:<\/p>\n

• John Baez, Klein’s quartic curve<\/a>.<\/p>\n

The image above is one among many generated by Anton Sherwood<\/a> using a Python program<\/a>. He put it in the public domain, and it is available on Wikicommons<\/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":"

This picture, drawn by Anton Sherwood, shows the {7,3} tiling<\/a><\/b>: a tiling of the hyperbolic plane by equal-sized regular heptagons, 3 meeting at each vertex. <\/p>\n

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