Smoothed octagon
The smoothed octagon is a region in the plane found by
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.[3] The following details apply to a regular octagon of circumradius with its centre at the point and one vertex at the point . For two constants and , the hyperbola is given by the equation
for the portion of the hyperbola that forms the corner, given by the range of parameter values
The lines of the octagon tangent to the hyperbola are , and the lines asymptotic to the hyperbola are simply .
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.[4] 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.[5] The smoothed octagon has a maximum packing density given by[2][3]
This is lower than the maximum packing density of circles, which is[3]
The maximum known packing density of the ordinary regular octagon is
Is the smoothed octagon the centrally symmetric shape with the lowest maximum packing density?
Reinhardt's conjecture that the smoothed octagon has the lowest maximum packing density of all centrally symmetric convex shapes in the plane remains unsolved. As a partial result,
If central symmetry is not required, the regular heptagon is conjectured to have even lower packing density, but neither its packing density nor its optimality have been proven. In three dimensions, Ulam's packing conjecture states that no convex shape has a lower maximum packing density than the ball.[5]
References
External links
- Packing smoothed octagons. John Baez, Visual Insight, AMS Blogs, 2014.
- The thinnest densest two-dimensional packing?. Peter Scholl, 2001.