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Smoothed octagon
Two-dimensional shape / From Wikipedia, the free encyclopedia
The smoothed octagon is a region in the plane found by Karl Reinhardt in 1934 and conjectured by him to have the lowest maximum packing density of the plane of all centrally symmetric convex shapes.[1] It was also independently discovered by Kurt Mahler in 1947.[2] It is constructed by replacing the corners of a regular octagon with a section of a hyperbola that is tangent to the two sides adjacent to the corner and asymptotic to the sides adjacent to these.
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![Thumb image](http://upload.wikimedia.org/wikipedia/commons/f/fc/SmoothedOctagonPackings.gif)