{"id":1135,"date":"2014-11-15T01:00:23","date_gmt":"2014-11-15T01:00:23","guid":{"rendered":"http:\/\/blogs.ams.org\/visualinsight\/?p=1135"},"modified":"2015-07-29T00:48:55","modified_gmt":"2015-07-29T00:48:55","slug":"packing-regular-heptagons","status":"publish","type":"post","link":"https:\/\/blogs.ams.org\/visualinsight\/2014\/11\/15\/packing-regular-heptagons\/","title":{"rendered":"Packing Regular Heptagons"},"content":{"rendered":"
\n
\"Densest<\/a>

Densest Double Lattice Packing of Regular Heptagons – Toby Hudson<\/p><\/div>\n<\/div>\n

This picture by Toby Hudson<\/a> shows the densest known packing of the regular heptagon. Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density.<\/p>\n

We listed some competitors in the last post:<\/p>\n

\u2022 Packing smoothed octagons<\/a>.<\/p>\n

The disk has a maximum packing density of:<\/p>\n

$$ \\frac{\\pi}{\\sqrt{12}} = 0.9068996 \\dots $$<\/p>\n

The regular octagon has a maximum packing density of:<\/p>\n

$$ \\frac{4 + 4 \\sqrt{2}}{5 + 4 \\sqrt{2}} = 0.9061636 \\dots $$<\/p>\n

If we smooth the corners of the regular octagon in a specific way, we get a maximum packing density of only:<\/p>\n

$$ \\frac{ 8-4\\sqrt{2}-\\ln{2} }{2\\sqrt{2}-1} = 0.902414 \\dots $$<\/p>\n

This is conjectured to be the lowest possible for a centrally symmetric<\/a> convex shape.<\/p>\n

But if we drop the constraint of central symmetry, the regular heptagon enters the game, and apparently beats the smoothed octagon! The densest known<\/i> packing of the regular heptagon, shown above, has density<\/p>\n

$$ \\frac{2}{97}\\left(-111 + 492 \\cos\\left(\\frac{\\pi}{7}\\right) – 356 \\cos^2 \\left(\\frac{\\pi}{7}\\right)\\right) = 0.89269 \\dots $$<\/p>\n

Greg Kuperberg and his father W\u0142odzimierz showed this is the densest ‘double-lattice packing’ of the regular heptagon. A lattice packing<\/b> of a shape is one in which all copies of that shape are translates of a fixed copy by lattice vectors. A double-lattice packing<\/b> of a shape is the union of two lattice packings such that a 180\u00b0 rotation about some point interchanges the two packings:<\/p>\n

\u2022 Greg Kuperberg and W\u0142odzimierz Kuperberg, Double-lattice packings of convex bodies in the plane<\/a>, Discrete and Computational Geometry<\/i> 5<\/b> (1990), 389–397.<\/p>\n

However, it has not been proved that this is the densest packing of the regular heptagon!<\/p>\n

The image here was created by Toby Hudson<\/a> and put on Wikicommons<\/a> with a Creative Commons Attribution-Share Alike 3.0 Unported<\/a> license.<\/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 by Toby Hudson<\/a> shows the densest known packing of the regular heptagon. Of all convex shapes, the regular heptagon is believed to have the lowest maximal packing density.<\/p>\n

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