Smooth Octagon

Harmonized octagon.

A family of maximally dense packages of a smoothed octagon.

A smoothed octagon is a region of the plane, presumably having the smallest highest packing density of the plane of all centrally symmetric convex figures ^[1] . The figure is obtained by replacing the angles of a regular octagon with a section of the hyperbola , which touches two sides of the corner and asymptotically approaches the extensions of the sides of the octagon, adjacent to the sides of the corner.

Maximum packing density

Smooth octagon has maximum packing density

{\frac {8-4{\sqrt {2}}-\ln {2}}{2{\sqrt {2}}-1}}\approx 0.902414\,.

{\ displaystyle {\ frac {8-4 {\ sqrt {2}} - \ ln {2}} {2 {\ sqrt {2}} - 1}} \ approx 0.902414 \ ,.}

{\frac {8-4{\sqrt {2}}-\ln {2}}{2{\sqrt {2}}-1}}\approx 0.902414\,.

^[2]

This density is less than the maximum packing density of the circles , which is equal to

{\frac {\pi }{\sqrt {12}}}\approx 0.906899.

{\ displaystyle {\ frac {\ pi} {\ sqrt {12}}} \ approx 0.906899.}

{\frac {\pi }{\sqrt {12}}}\approx 0.906899.

The maximum packing density of regular regular octagons is

{\frac {4+4{\sqrt {2}}}{5+4{\sqrt {2}}}}\approx 0.906163,

{\ displaystyle {\ frac {4 + 4 {\ sqrt {2}}} {5 + 4 {\ sqrt {2}}}} \ approx 0.906163,}

{\frac {4+4{\sqrt {2}}}{5+4{\sqrt {2}}}}\approx 0.906163,

which is also slightly less than the maximum packing density of the circles, but more than the packing density of a smoothed octagon ^[3] .

The smoothed octagon reaches its maximum packing density not only for a single package, but for a one-parameter family of packages. All of them are lattice packaging ^[4] .

For three-dimensional space states that there is no convex figure with the highest packing density less than the packing of balls.

Build

The corners of a smoothed octagon can be found by rotating three regular octagons, the centers of which form a triangle with a constant area.

When considering families of maximally dense packings of a smoothed octagon, the requirement that the packing density remains unchanged when the contact point of adjacent octagons changes can be used to determine the shape of the corners. In the figure, the three octagons rotate, while the area of the triangle formed by the centers of these octagons does not change. For regular octagons, the red and blue fragments overlap, so to rotate it is necessary to cut off the angles at a point halfway between the centers of the octagons, which gives a curve that turns out to be a hyperbola.

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

The hyperbola is constructed as a tangent to two sides of the octagon and having asymptotes two sides of the octagon adjacent to these sides. Arrange a regular octagon with the radius of the circumscribed circle. ${\sqrt {2}}$ ${\ displaystyle {\ sqrt {2}}}$ and center at $(2+{\sqrt {2}},0)$ ${\ displaystyle (2 + {\ sqrt {2}}, 0)}$ . One vertex will be at a point $(2,0)$ ${\ displaystyle (2.0)}$ . Define two constants, ℓ and m :

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

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

m={\sqrt[{4}]{\frac {1}{2}}}

{\ displaystyle m = {\ sqrt [{4}] {\ frac {1} {2}}}}

Hyperbola is then given by the equation

\ell ^{2}x^{2}-y^{2}=m^{2}

{\ displaystyle \ ell ^ {2} x ^ {2} -y ^ {2} = m ^ {2}}

or, in equivalent parameterized form (only for the right side of the hyperbola):

{\begin{cases}x={\frac {m}{\ell }}\cosh {t}\\y=m\sinh {t}\end{cases}}-\infty <t<\infty

{\ displaystyle {\ begin {cases} x = {\ frac {m} {\ ell}} \ cosh {t} \\ y = m \ sinh {t} \ end {cases}} - \ infty <t <\ infty}

The portion of the hyperbola that forms the corners is determined by the parameter values

-{\frac {\ln {2}}{4}}<t<{\frac {\ln {2}}{4}}

{\ displaystyle - {\ frac {\ ln {2}} {4}} <t <{\ frac {\ ln {2}} {4}}}

Octagon lines tangent to hyperbole

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

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

Asymptotes for hyperbole

y=\pm \ell x.

{\ displaystyle y = \ pm \ ell x.}

Notes

↑ Reinhardt, 1934 , p. 216-230.
↑ Weisstein, Eric W. Smoothed Octagon on the Wolfram MathWorld website.
↑ Atkinson, Jiao, Torquato, 2012 .
↑ Kallus, 2013 .

Literature

K. Reinhardt. Über die dichteste gitterförmige Lagerung kongruenter Bereiche in der Ebene und eine besondere Art konvexer Kurven // Abh. Math. Sem. Hamburg - 1934. - Issue. 10 . - S. 216-230 .
Steven Atkinson, Yang Jiao, Salvatore Torquato. Maximally dense packings of two-dimensional convex and concave noncircular particles // Phys. Rev. E. - 2012 .-- T. 86 , no. 03 . Archived August 24, 2014.
Yoav Kallus. Least efficient packing shapes // Geometry and Topology. - 2013. - May.

Links

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

[_137dad5da69a1a7c-1] Reinhardt, 1934 , p. 216-230.

[2] Weisstein, Eric W. Smoothed Octagon on the Wolfram MathWorld website.

[_a762a2f7a64200bf-3] Atkinson, Jiao, Torquato, 2012 .

[_aed488f83ad93783-4] Kallus, 2013 .