Ball Packings Research Articles

Numerical simulation and experimental investigations of hydrodynamics and heat transfer for radial gas flow in an apparatus with ball packing

Results of numerical simulation and experimental data on hydrodynamics and heat transfer during radial gas flow in a bed of ball elements are presented.

A Continuous Bijection from $\ell^{2}$ Onto a Subset of $\ell^{2}$ Whose Inverse is Everywhere Unboundedly Discontinuous, with an Application to Packing of Balls in $\ell^2$

There is a continuous bijection from $ \ell^{2}$ onto a subset of $\ell^{2}$ whose inverse is everywhere unboundedly discontinuous. If $B$ is a ball in $\ell^2$, then the continuous bijection defined on $\ell^2$ maps countably many mutually disjoint balls of $\ell^2$ into countably many mutually disjoint balls in $B$, making those images mutually disjoint.

Maximal Ball Packings of Symplectic-Toric Manifolds

Let (M, σ, ψ) be a symplectic-toric manifold of dimension at least four. This article investigates the symplectic ball packing problem in the toral equivariant setting. We show that the set of toric symplectic ball packings of M admits the structure of a convex polytope. Previous work of the first author shows that up to equivalence, only (ℂℙ1)2 and ℂℙ2 admit density one packings when n = 2 and only ℂℙn admits density one packings when n > 2. In contrast, we show that for a fixed n ≥ 2 and each δ ∈ (0, 1), there are uncountably many genuinely inequivalent 2n-dimensional symplectic-toric manifolds with a maximal toric packing of density δ. This result follows from a general analysis of how the densities of maximal packings change while varying a given symplectic-toric manifold through a family of symplectic-toric manifolds that are equivariantly diffeomorphic but not equivariantly symplectomorphic.

Publisher's Note: Model of random packings of different size balls [Phys. Rev. E81, 051303 (2010)

Publisher's Note: Model of random packings of different size balls [Phys. Rev. E<b>81</b>, 051303 (2010)

Model of random packings of different size balls

We develop a model to describe the properties of random assemblies of polydisperse hard spheres. We show that the key features to describe the system are (i) the dependence between the free volume of a sphere and the various coordination numbers between the species and (ii) the dependence of the coordination numbers with the concentration of species; quantities that are calculated analytically. The model predicts the density of random close packing and random loose packing of polydisperse systems for a given distribution of ball size and describes packings for any interparticle friction coefficient. The formalism allows to determine the optimal packing over different distributions and may help to treat packing problems of nonspherical particles which are notoriously difficult to solve.

The dodecahedral conjecture

This article gives a summary of a proof of Fejes Toth’s Dodecahedral conjecture: the volume of a Voronoi polyhedron in a three-dimensional packing of balls of unit radius is at least the volume of a regular dodecahedron of unit inradius.

Simultaneous Packing and Covering in Sequence Spaces

We adapt a construction of Klee (1981) to find a packing of unit balls in $\ell_p$ ($1\leq p<\infty$) which is efficient in the sense that enlarging the radius of each ball to any $R>2^{1-1/p}$ covers the whole space. We show that the value $2^{1-1/p}$ is optimal.

A Revision of the Proof of the Kepler Conjecture

The Kepler conjecture asserts that no packing of congruent balls in three-dimensional Euclidean space has density greater than that of the face-centered cubic packing. The original proof, announced in 1998 and published in 2006, is long and complex. The process of revision and review did not end with the publication of the proof. This article summarizes the current status of a long-term initiative to reorganize the original proof into a more transparent form and to provide a greater level of certification of the correctness of the computer code and other details of the proof. A final part of this article lists errata in the original proof of the Kepler conjecture.

Random thoughts on densest packing

Chaikin replies: The readers’ comments remind me of occasions when I’ve run over the time limit while giving a talk, and the first question is, “What would you say if you had another 10 minutes?”In response to David Rosen, I offer the following. Several infinite structures have the same 0.74 packing fraction as the face-centered cubic structure. In referring to FCC, I was following the spirit of Thomas Hales in his 2005 paper on the proof of the densest packing in three dimensions. 1 1. T. C. Hales, Ann. Math. 162, 1065 (2005). https://doi.org/10.4007/annals.2005.162.1065 According to his theorem 1.1, the Kepler conjecture, no packing of congruent balls in Euclidean three space has density greater than that of the FCC packing.Equivalently high packings are found by stacking planes of hexagonal close-packed spheres. Each sheet of spheres is placed so that it sits in the interstices of the previous hexagonal sheet. There are two sets of interstitial sites: B and C; the first layer is called A. The following are some common stackings: ABCABCABC, the FCC structure; ABABABABAB, hexagonal close packed or HCP; and ABACABAC, double HCP, a structure found for some lanthanide and actinide elements. And ABCBACBCBACABCA is random HCP, the structure found in many colloidal crystals. However, the number of such configurations grows exponentially with the length of a sample rather than its volume and thus does not contribute to the sample’s entropy. I also found Terry Goldman’s comments interesting. Strangely enough, several research groups, including my own, have performed crystallization experiments on colloids in micro-gravity during orbit. The physics of the liquid-to-crystal transition is the same in space as on the ground. That is because the transition has its basis in the purely geometric problem of particle packing. Using the crystal and random packing limits that researchers have found in experiments on granular systems, we can evaluate the entropy of a system at lower densities, and the ordered state has the higher entropy. For molecules on Earth and for colloids in space, thermal energy dominates gravitational potential energy, and entropic effects produce the crystallization with which we are familiar.REFERENCESection:ChooseTop of pageREFERENCE <<1. T. C. Hales, Ann. Math. 162, 1065 (2005). https://doi.org/10.4007/annals.2005.162.1065 , Google ScholarCrossref© 2008 American Institute of Physics.

Distinctive features of a crystal, crystal-like properties of a liquid and atomic quantum effects

It is believed that ‘a crystal is similar to the crowd which is tightly compressed within enclosed space’ and its structure in the simplest case is similar to the closest ball packing. Based on this assumption the strength of a crystal, long range ordering, the granular structure, capability for polymorphic transformation etc. were deduced. In a liquid such properties are impossible even in feebly marked form. However some of crystal-like features of melts are revealed in experiments and they frequently remain unacknowledged with a theory. From the other hand, computer model of crystal does not give even listed distinctive features of a crystal state. In the classical model the solidification more than to sunflower oil consistence was not obtained. It is possible to reach the real solidification if quantum ‘freezing’ of a part of atomic degrees of freedom would taken into account and any movement would stopped at zero energy level. There are some reasons to believe that another crystal properties and corresponding crystal-like features of liquids also can be got basing on these atomic quantum effects. In this case the reasons of many discussions on ‘heredity’, ‘memory’ of liquid and its microheterogeneity disappear.

Hydraulic resistance of bulk metallic-ball packing of regenerator of air-fractionating plant

The principal ways of mathematically describing boundaries of bulk (loose) ball packings are discussed. A model of the typical component of the ball packing and a procedure for determining the hydraulic resistance of the packing are proposed. The proposed method helps calculate the hydraulic resistance of packings of various shapes for selecting the optimum parameters.

A new lower bound on the surface area of a Voronoi polyhedron

We prove that the mininum surface area of a Voronoi cell in a unit ball packing in ]]> ]]> ]]> ]]> ]]> {\mathbb E}^3$ is at least $16.1977$. This result provides further support for the Strong Dodecahedral Conjecture according to which the minimum surface area of a Voronoi cell in a $3$-dimensional unit ball packing is at least as large as the surface area of a regular dodecahedron of inradius $1$, which is about $16.6508\ldots\,$.

Toric symplectic ball packing

We define and solve the toric version of the symplectic ball packing problem, in the sense of listing all 2 n-dimensional symplectic–toric manifolds which admit a perfect packing by balls embedded in a symplectic and torus equivariant fashion. In order to do this we first describe a problem in geometric–combinatorics which is equivalent to the toric symplectic ball packing problem. Then we solve this problem using arguments from Convex Geometry and Delzant theory. Applications to symplectic blowing-up are also presented, and some further questions are raised in the last section.

Zhdanov's rules work both ways

The relationship between the space-group symmetry of a close packing of equal balls of repeat period P and the symmetry properties of its representing Zhdanov symbol is analyzed. Proofs are straightforward when some symmetry is assumed for the stacking, and it is investigated how this symmetry is reflected in the structure of the Zhdanov symbol. Most of these proofs are documented in the literature, with variable degrees of rigor. However, the proof is somewhat more involved when working backwards, i.e. when some symmetry properties for the Zhdanov symbol are assumed and the corresponding effect on the symmetry of the polytype structure it represents is investigated, which may explain why these proofs are avoided or shrugged off as ;easily seen', 'obvious' and the like.

Sphere Packing, IV. Detailed Bounds

This paper is the fourth in a series of six papers devoted to the proof of theKepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In a previous paper in this series, a continuous function f on a compact spacewas defined, certain points in the domain were conjectured to give the global maxima, and the relation between this conjecture and the Kepler conjecture was established. The function f can be expressed as a sum of terms, indexed by regions on a unit sphere. In this paper detailed estimates of the terms corresponding to general regions are developed. These results form the technical heart of the proof of the Kepler conjecture, by giving detailed bounds on the function f. The results rely on long computer calculations.

A Formulation of the Kepler Conjecture

This paper is the second in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. The top level structure of the proof is described. A compact topological space is described. Each point of this space can be described as a finite cluster of balls with additional combinatorial markings. A continuous function on this compact space is defined. It is proved that the Kepler conjecture will follow if the value of this function is never greater than a given explicit constant.

Sphere Packing, III. Extremal Cases

This paper is the third in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In the previous paper in this series, a continuous function f on a compact space was defined, certain points in the domain were conjectured to give the global maxima, and the relation between this conjecture and the Kepler conjecture was established. This paper shows that those points are indeed local maxima. Various approximations to f are developed, that will be used in subsequent papers to bound the value of the function f. The function f can be expressed as a sum of terms, indexed by regions on a unit sphere. Detailed estimates of the terms corresponding to triangular and quadrilateral regions are developed.

Historical Overview of the Kepler Conjecture

This paper is the first in a series of six papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. After some preliminary comments about the face-centered cubic and hexagonal close packings, the history of the Kepler problem is described, including a discussion of various published bounds on the density of sphere packings. There is also a general historical discussion of various proof strategies that have been tried with this problem.

Sphere Packings, VI. Tame Graphs and Linear Programs

This paper is the sixth and final part in a series of papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In a previous paper in this series, a continuous function f on a compact space is defined, certain points in the domain are conjectured to give the global maxima, and the relation between this conjecture and the Kepler conjecture is established. In this paper we consider the set of all points in the domain for which the value of f is at least the conjectured maximum. To each such point, we attach a planar graph. It is proved that each such graph must be isomorphic to a tame graph, of which there are only finitely many up to isomorphism. Linear programming methods are then used to eliminate all possibilities, except for three special cases treated in earlier papers: pentahedral prisms, the face-centered cubic packing, and the hexagonal-close packing. The results of this paper rely on long computer calculations.

Sphere Packings, V. Pentahedral Prisms

This paper is the fifth in a series of papers devoted to the proof of the Kepler conjecture, which asserts that no packing of congruent balls in three dimensions has density greater than the face-centered cubic packing. In this paper we prove that decomposition stars associated with the plane graph of arrangements we term pentahedral prisms do not contravene. Recall that a contravening decomposition star is a potential counterexample to the Kepler conjecture. We use interval arithmetic methods to prove particular linear relations on components of any such contravening decomposition star. These relations are then combined to prove that no such contravening stars exist.