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
(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.
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,
Hamburg10, 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."
