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Smoothed Octagon


A plane shape constructed by Reinhardt (1934) that is conjectured to be the "worst" packer of all centrally-symmetric plane regions. It has a packing density of

 eta=(8-4sqrt(2)-ln2)/(2sqrt(2)-1)=0.902414...

(OEIS A093767), significantly less than the circle packing density of

 eta=pi/(sqrt(12))=0.906899...

(OEIS A093766). The smoothed octagon is constructed from a regular octagon by smoothing the edges using a hyperbola that is tangent to adjacent edges of the octagon and has the edges adjacent to these as asymptotes.


See also

Circle Packing, Octagon

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References

Fejes Tóth, G. Lagerungen in der Ebene, auf der Kugel und in Raum, 2nd ed. Berlin: Springer-Verlag, p. 104, 1972.Fejes Toth, G. and Kuperberg, W. "Packing and Covering with Convex Sets." §3.3 in Handbook of Convex Geometry (Ed. P. M. Gruber and J. M. Wills). Amsterdam, Netherlands: North-Holland, pp. 799-860, 1993.Pach, J. and Agarwal, P. K. Combinatorial Geometry. New York: Wiley, p. 30, 1995.Reinhardt, K. "Über die dichteste gitterförmige Lagerung kongruente Bereiche in der Ebene und eine besondere Art konvexer Kurven." Abh. Math. Sem., Hamburg, Hansischer Universität, Hamburg 10, 216-230, 1934.Scholl, P. "The Thinnest Densest Two-Dimensional Packing?" http://www.home.unix-ag.org/scholl/octagon.html.Sloane, N. J. A. Sequences A093766 and A093767 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Smoothed Octagon

Cite this as:

Weisstein, Eric W. "Smoothed Octagon." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SmoothedOctagon.html

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