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).
The animation above is adapted from code supplied by E. Pegg, Jr. (pers. comm., 2026), based on the Smoothed Octagon Playground (Vu-Le). It shows a continuous
family of packings with the central octagon fixed. All
copies keep the same orientation while the point lattice
of their centers changes. The six marked contact points
form a centrally symmetrichexagon
of constant area, and the packing
density remains unchanged (Hales 2011).
