Smoothed octagon

A smoothed octagon.

The best known packing of smoothed octagons.

The smoothed octagon is a geometrical construction conjectured to have the lowest maximum packing density of the plane of all centrally symmetric convex shapes. 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.

The smoothed octagon has a maximum packing density, η_so given by

$\eta_{so} = \frac{ 8-4\sqrt{2}-\ln{2} }{2\sqrt{2}-1} \approx 0.902414 \, .$

This is lower than the maximum packing density of circles, which is

$\frac{\pi}{\sqrt{12}} \approx 0.9069.$

Construction

Construction of the smoothed octagon (black), the tangent hyperbola (red) and the asymptotes of this hyperbola (green), and the tangent sides to the hyperbola (blue).

The hyperbola is constructed tangent to two sides of the octagon, and asymptotic to the two adjacent to these. If we define two constants, ℓ and m:

$\ell = \sqrt{2} - 1$

$m = \sqrt{ 6 \sqrt{2} - 8} \frac{\sqrt{2}+1}{2}$

The hyperbola is then given by the equation

$\ell^2x^2-y^2=m^2$

or the equivalent parametrisation (for the right-hand branch only):

$x=\frac{m}{\ell} \cosh{t}; \quad y = m \sinh t ; \quad -\pi<t<\pi$

The lines of the octagon tangent to the hyperbola are

$y= \pm \left(\sqrt{2} + 1 \right) \left( x-2 \right)$

The lines asymptotic to the hyperbola are simply

$y = \pm \ell x.$

See also

• Circle packing

External links

• The thinnest densest two-dimensional packing?. Peter Scholl, 2001.