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

Hyperbola

In mathematics a hyperbola is a curve, specifically a smooth curve that lies in a plane, which can be defined either by its geometric properties or by the kinds of equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, which are mirror...

 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

This is lower than the maximum packing density of circles

Circle packing

In geometry, circle packing is the study of the arrangement of circles on a given surface such that no overlapping occurs and so that all circles touch another. The associated "packing density", η of an arrangement is the proportion of the surface covered by the circles...

, which is

Construction

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:



The hyperbola is then given by the equation


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

The lines of the octagon tangent to the hyperbola are


The lines asymptotic to the hyperbola are simply

External links

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