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. It was also independently discovered by Kurt Mahler in 1947. 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.
Construction
The hyperbola that forms each corner of the smoothed octagon is tangent to two sides of a regular octagon, and asymptotic to the two adjacent to these. The following details apply to a regular octagon of circumradius 2 {\displaystyle {\sqrt {2}}} with its centre at the point ( 2 + 2 , 0 ) {\displaystyle (2+{\sqrt {2}},0)} and one vertex at the point ( 2 , 0 ) {\displaystyle (2,0)} . For two constants ℓ = 2 − 1 {\displaystyle \ell ={\sqrt {2}}-1} and m = ( 1 / 2 ) 1 / 4 {\displaystyle m=(1/2)^{1/4}} , the hyperbola is given by the equation
ℓ 2 x 2 − y 2 = m 2 {\displaystyle \ell ^{2}x^{2}-y^{2}=m^{2}}
or the equivalent parameterization (for the right-hand branch only)
x = m ℓ cosh t y = m sinh t {\displaystyle {\begin{aligned}x&={\frac {m}{\ell }}\cosh {t}\\y&=m\sinh {t}\\\end{aligned}}}
for the portion of the hyperbola that forms the corner, given by the range of parameter values
− ln 2 4 < t < ln 2 4 . {\displaystyle -{\frac {\ln {2}}{4}}<t<{\frac {\ln {2}}{4}}.}
The lines of the octagon tangent to the hyperbola are y = ± ( 2 + 1 ) ( x − 2 ) {\displaystyle y=\pm \left({\sqrt {2}}+1\right)\left(x-2\right)} , and the lines asymptotic to the hyperbola are simply y = ± ℓ x {\displaystyle y=\pm \ell x} .
Packing
For every centrally symmetric convex planar set, including the smoothed octagon, the maximum packing density is achieved by a lattice packing, in which unrotated copies of the shape are translated by the vectors of a lattice. The smoothed octagon achieves its maximum packing density, not just for a single packing, but for a 1-parameter family. All of these are lattice packings. The smoothed octagon has a maximum packing density given by
8 − 4 2 − ln 2 2 2 − 1 ≈ 0.902414 . {\displaystyle {\frac {8-4{\sqrt {2}}-\ln {2}}{2{\sqrt {2}}-1}}\approx 0.902414\,.}
This is lower than the maximum packing density of circles, which is
π 12 ≈ 0.906899. {\displaystyle {\frac {\pi }{\sqrt {12}}}\approx 0.906899.}
The maximum known packing density of the ordinary regular octagon is
4 + 4 2 5 + 4 2 ≈ 0.906163 , {\displaystyle {\frac {4+4{\sqrt {2}}}{5+4{\sqrt {2}}}}\approx 0.906163,}
also slightly less than the maximum packing density of circles, but higher than that of the smoothed octagon.
Conjectured optimality
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