The smoothed octagon is a region in the plane found by Karl Reinhardt in 1934 and conjectured by him to have the ''lowest'' maximum

packing density A packing density or packing fraction of a packing in some space is the fraction of the space filled by the figures making up the packing. In simplest terms, this is the ratio of the volume of bodies in a space to the volume of the space itself. ...

of the plane of all

centrally symmetric In geometry, a point reflection (also called a point inversion or central inversion) is a geometric transformation of affine space in which every point is reflected across a designated inversion center, which remains fixed. In Euclidean or ...

convex shapes. It was also independently discovered by

Kurt Mahler Kurt Mahler FRS (26 July 1903 – 25 February 1988) was a German mathematician who worked in the fields of transcendental number theory, diophantine approximation, ''p''-adic analysis, and the geometry of numbers.

in 1947. It is constructed by replacing the corners of a regular octagon with a section of a

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

that is tangent to the two sides adjacent to the corner and asymptotic to the sides adjacent to these.

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. The following details apply to a regular octagon of circumradius

\sqrt

with its centre at the point

(2+\sqrt,0)

and one vertex at the point

(2,0)

. For two constants

\ell=\sqrt - 1

and

m=(1/2)^

, the hyperbola is given by the equation

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

or the equivalent parameterization (for the right-hand branch only)

\begin
x&=\frac \cosh\\
y&= m \sinh\\
\end

for the portion of the hyperbola that forms the corner, given by the range of parameter values

-\frac The lines of the octagon tangent to the hyperbola are y= \pm \left(\sqrt + 1 \right) \left( x-2 \right),
and the lines asymptotic to the hyperbola are simply y = \pm \ell x .

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. 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. The smoothed octagon has a maximum packing density given by

\frac \approx 0.902414 \, .

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

\frac \approx 0.906899.

The maximum known packing density of the ordinary regular octagon is

\frac \approx 0.906163,

also slightly less than the maximum packing density of circles, but higher than that of the smoothed octagon.

Conjectured optimality

Reinhardt's conjecture that the smoothed octagon has the lowest maximum packing density of all centrally symmetric convex shapes in the plane remains unsolved. However, Thomas Hales and Koundinya Vajjha claimed to have proved a weaker conjecture, which asserts that the most unpackable centrally symmetric convex disk must be a smoothed polygon. Additionally, Fedor Nazarov provided a partial result by proving that the smoothed octagon is a

local minimum In mathematical analysis, the maximum and minimum of a function are, respectively, the greatest and least value taken by the function. Known generically as extremum, they may be defined either within a given range (the ''local'' or ''relative ...

for packing density among centrally symmetric convex shapes. If central symmetry is not required, the regular

heptagon In geometry, a heptagon or septagon is a seven-sided polygon or 7-gon. The heptagon is sometimes referred to as the septagon, using ''Wikt:septa-, septa-'' (an elision of ''Wikt:septua-, septua-''), a Latin-derived numerical prefix, rather than ...

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.

References

{{Reflist, refs= {{cite journal , last1 = Atkinson , first1 = Steven , last2 = Jiao , first2 = Yang , last3 = Torquato , first3 = Salvatore , arxiv = 1405.0245 , bibcode = 2012PhRvE..86c1302A , date = September 10, 2012 , doi = 10.1103/physreve.86.031302 , issue = 3 , journal = Physical Review E , page = 031302 , pmid = 23030907 , s2cid = 9806947 , title = Maximally dense packings of two-dimensional convex and concave noncircular particles , volume = 86 {{cite journal , last = Fejes Tóth , first = László , author-link = László Fejes Tóth , journal = Acta Universitatis Szegediensis , mr = 38086 , pages = 62–67 , title = Some packing and covering theorems , volume = 12 , year = 1950 {{cite journal , last = Kallus , first = Yoav , arxiv = 1305.0289 , doi = 10.2140/gt.2015.19.343 , issue = 1 , journal = Geometry & Topology , mr = 3318753 , pages = 343–363 , title = Pessimal packing shapes , volume = 19 , year = 2015 {{cite book , last1 = Fejes Tóth , first1 = László , author1-link = László Fejes Tóth , last2 = Fejes Tóth , first2 = Gábor , last3 = Kuperberg , first3 = Włodzimierz , author3-link = Włodzimierz Kuperberg , doi = 10.1007/978-3-031-21800-2 , isbn = 978-3-031-21799-9 , location = Cham , mr = 4628019 , page
106
, publisher = Springer , series = Grundlehren der mathematischen Wissenschaften undamental Principles of Mathematical Sciences , title = Lagerungen: Arrangements in the Plane, on the Sphere, and in Space , volume = 360 , year = 2023 {{cite journal , last = Mahler , first = Kurt , author-link = Kurt Mahler , journal =

Indagationes Mathematicae ''Indagationes Mathematicae'' (Latin for "Mathematical Investigations") is a Dutch mathematics journal. The journal originates from the ''Proceedings of the Royal Netherlands Academy of Arts and Sciences'' (or ''Proceedings of the Koninklijke N ...

, mr = 21017 , pages = 326–337 , title = On the minimum determinant and the circumscribed hexagons of a convex domain , url = https://dwc.knaw.nl/DL/publications/PU00018367.pdf , volume = 9 , year = 1947 {{cite journal , last = Nazarov , first = F. L. , author-link = Fedor Nazarov , doi = 10.1007/BF01727653 , journal = Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta imeni V. A. Steklova Akademii Nauk SSSR (LOMI) , mr = 849319 , pages = 104–114, 197–198 , title = On the Reinhardt problem of lattice packings of convex regions: Local extremality of the Reinhardt octagon , volume = 151 , year = 1986 {{cite journal , last = Reinhardt , first = Karl , author-link = Karl Reinhardt (mathematician) , doi = 10.1007/BF02940676 , journal = Abh. Math. Sem. Univ. Hamburg , pages = 216–230 , s2cid = 120336230 , title = Über die dichteste gitterförmige Lagerung kongruenter Bereiche in der Ebene und eine besondere Art konvexer Kurven , volume = 10 , year = 1934

External links

Packing smoothed octagons
John Baez, Visual Insight, AMS Blogs, 2014.

Peter Scholl, 2001. Packing problems