A smoothed octagon is a plane shape constructed by Reinhardt (1934) that is conjectured to be the "worst" packer of all centrally symmetric
convex bodies in the plane.
It has a packing density of
(OEIS A093767; Hales 2011, p. 2), significantly
less than the circle packing density of
(OEIS A093766). The smoothed octagon is constructed from a regular octagon by rounding the corners
using a hyperbola that is tangent to adjacent edges
of the octagon and has the edges adjacent to these as
asymptotes (Hales 2011, p. 2).
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.Fejes
Tóth, L. Lagerungen
in der Ebene, auf der Kugel und im Raum, 2nd ed. Berlin, Germany: Springer-Verlag,
p. 104, 1972.Hales, T. C. "On the Reinhardt Conjecture."
23 Mar 2011. https://arxiv.org/abs/1103.4518.Pach,
J. and Agarwal, P. K. Combinatorial
Geometry. New York: Wiley, p. 30, 1995.Reinhardt, K. "Über
die dichteste gitterförmige Lagerung kongruenter Bereiche in der Ebene und eine
besondere Art konvexer Kurven." Abh. Math. Sem., Hamburg, Hansischer Universität,
Hamburg10, 216-230, 1934. https://doi.org/10.1007/BF02940676.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."
