Spheres and sausages, crystals and catastrophes -- and a joint packing theory (Q1272417)

This paper gives an introduction to the theory of finite packings which are ``free'' in the sense that no container is required. The author asks the following seven questions (and answers them): Which packing of four tennis balls is denser: the tetrahedron or the linear one? The same question for 56 balls. Is the linear packing the densest packing of five balls? What is the densest packing of \(n\) billiard balls on a table? How can one compare the density of two crystals of the same structure but different shape, say of two diamonds? And what is the densest packing of identical coins on a table -- and in space? If \(C_n= [c_1,c_2,\dots, c_n]\) is the set of centers of \(n\) balls (in a packing) then the packing is called a sausage when \(\dim(\text{conv}(C_n))= 1\), a pizza when \(2\leq \dim(\text{conv}(C_n))\leq d-1\) and a cluster when \(\dim(\text{conv}(C_n))= d\). So why the 56 in the second problem? Because in \(E^3\) the densest packing of \(n\) balls is (known to be) a cluster if \(n=56\) and \(59\leq n\leq 62\), or \(n\geq 65\). It is conjectured that for all other \(n\), the sausage is optimal. So under this assumption, the shape of the densest ball packing changes drastically from \(n\leq 55\) to \(n\geq 56\). The first four sections in the paper are titled: Seven Questions; Finite Packings; Densities; Sausages, Pizzas, Clusters, and Catastrophes. There are many strange phenomena illuminated and they show that a joint theory of finite and infinite packings cannot be expected. The crucial idea for overcoming these difficulties is explained in the section titled Parametric Density, whose basic properties are given in the section Basic Properties. This leads to section The Joint Packing Theory. The final section deals with Applications to Crystal Growth. Throughout the paper there are exceptionally fine diagrams and visualizations, all computergenerated by a 16 year old student! No review can do justice to a paper which is only 6 pages long but nevertheless contains such a fine overview of a (joint) packing theory. Recommended reading for anyone interested in this branch of geometry.