Finite sphere packing
The question of packing finitely many spheres has only been investigated in detail in recent decades, with much of the groundwork being laid by László Fejes Tóth.
The similar problem for infinitely many spheres has a longer history of investigation, from which the Kepler conjecture is most well-known.
In general, a packing refers to any arrangement of a set of spatially-connected, possibly differently-sized or differently-shaped objects in space such that none of them overlap.
Such a packing of spheres determines a specific volume known as the convex hull of the packing, defined as the smallest convex set that includes all the spheres.
There are many possible ways to arrange spheres, which can be classified into three basic groups: sausage, pizza, and cluster packing.
[1]An arrangement in which the midpoint of all the spheres lie on a single straight line is called a sausage packing, as the convex hull has a sausage-like shape.
An approximate example in real life is the packing of tennis balls in a tube, though the ends must be rounded for the tube to coincide with the actual convex hull.
Real-life approximations include fruit being packed in multiple layers in a box.
dimensions, "sausages" refer to one-dimensional arrangements, "clusters" to
The empty space between spheres varies depending on the type of packing.
[1] The sudden transition in optimal packing shape is jokingly known by some mathematicians as the sausage catastrophe (Wills, 1985).
dimensions the sudden transition is conjectured to happen around 377000 spheres.
This result only concerns spheres and not other convex bodies; in fact Gritzmann and Arhelger observed that for any dimension
Similarly, it is possible to find the volume of the convex hull of a tetrahedral packing, in which the spheres are arranged so that they form a tetrahedral shape, which only leads to completely filled tetrahedra for specific numbers of spheres.
It is also possible with some more effort to derive the exact formula for the volume of the tetrahedral convex hull
, which would involve subtracting the excess volume at the corners and edges of the tetrahedron.
This allows the sausage packing to be proved non-optimal for smaller values of
The term sausage comes from the mathematician László Fejes Tóth, who posited the sausage conjecture in 1975,[5] which concerns a generalized version of the problem to spheres, convex hulls, and volume in higher dimensions.
upwards it is always optimal to arrange the spheres along a straight line.
The best results so far are those of Ulrich Betke und Martin Henk,[6] who proved the conjecture for dimensions 42 and above.
It is difficult to find the optimal packing as there is no "simple" formula for the volume of an arbitrarily shaped cluster.
Optimality (and non-optimality) is shown through appropriate estimates of the volume, using methods from convex geometry, such as the Brunn-Minkowski inequality, mixed Minkowski volumes and Steiner's formula.
A crucial step towards a unified theory of both finite and infinite (lattice and non-lattice) sphere packings was the introduction of parametric densities by Jörg Wills in 1992.
The parametric density takes into account the influence of the edges of the packing.
For a linear arrangement (sausage), the convex hull is a line segment through all the midpoints of the spheres.
This definition works in two dimensions, where Laszlo Fejes-Toth, Claude Rogers and others used it to formulate a unified theory of finite and infinite packings.
In three dimensions, Wills gives a simple argument that such a unified theory is not possible based on this definition: The densest finite arrangement of coins in three dimensions is the sausage with
To solve this issue, Wills introduces a modification to the definition by adding a positive parameter
allows the influence of the edges to be considered (giving the convex hull a certain thickness).
This is then combined with methods from the theory of mixed volumes and geometry of numbers by Hermann Minkowski.
