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[[File:Smoothed Octagon Simple.svg||thumb|150px|A smoothed octagon.]]
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[[File:Smoothed Octagon Packed.svg||thumb|150px|The best packing of smoothed octagons.]]
The '''smoothed octagon''' is a geometrical construction conjectured to have the ''lowest'' maximum [[packing density]] of the [[Plane (geometry)|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''<sub>so</sub> given by
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:<math>\eta_{so} = \frac{ 8-4\sqrt{2}-\ln{2} }{2\sqrt{2}-1} \approx 0.902414 \, .</math><ref>{{MathWorld|urlname=SmoothedOctagon |title=Smoothed Octagon }}</ref>
 
This is lower than the [[circle packing|maximum packing density of circles]], which is
 
:<math>\frac{\pi}{\sqrt{12}} \approx 0.9069.</math>
 
==Construction==
[[File:Smoothed Octagon.svg|thumb|400px|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'':
 
:<math>\ell = \sqrt{2} - 1  </math>
 
:<math>m = \sqrt{ 6 \sqrt{2} - 8} \frac{\sqrt{2}+1}{2}</math>
 
The hyperbola is then given by the equation
 
:<math>\ell^2x^2-y^2=m^2 </math>
 
or the equivalent parameterization (for the right-hand branch only):
 
: <math>x=\frac{m}{\ell} \cosh{t}; \quad y = m \sinh t ; \quad -\pi<t<\pi</math>
 
The lines of the octagon tangent to the hyperbola are
 
:<math>y= \pm \left(\sqrt{2} + 1 \right) \left( x-2 \right) </math>
 
The lines asymptotic to the hyperbola are simply
 
:<math>y = \pm \ell x.  </math>
 
==See also==
*[[Circle packing]]
 
==References==
{{Reflist}}
 
==External links==
*[http://www.home.unix-ag.org/scholl/octagon.html The thinnest densest two-dimensional packing?]. Peter Scholl, 2001.
 
[[Category:Tessellation]]

Latest revision as of 19:07, 5 June 2014

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