Ball Packings Research Articles

Optimal ball packings for crystallographic groups of the cubic crystal system and their Dirichlet-Voronoi cells

Abstract In this paper we look for densest ball packings of Euclidean space E3 to given symmetry groups. We restrict our investigation to the 36 space groups of the cubic crystal system, and we search for only those packings where the group acts simply transitively on the balls. In order to find the centre of a ball and its radius for the case of an optimal packing we will apply an algorithm and the corresponding computer program, which was developed by the second author [21, 12]. In the list of our results we will give the coordinates of the ball centre and the radius, moreover, we will compute the density of the optimal packing and display the corresponding D-V cell for each space group above (see also [14, 15]).

Lattices of Optimal Finite Lattice Packings

We consider finite lattice ball packings with respect to parametric density and show that densest packings are attained in critical lattices if the number of translates and the density parameter are sufficiently large. A corresponding result is not valid for general centrally symmetric convex bodies.

Efficiency of various lattices from hard ball to soft ball: theoretical study of thermodynamic properties of dendrimer liquid crystal from atomistic simulation.

Self-assembled supramolecular organic liquid crystal structures at nanoscale have potential applications in molecular electronics, photonics, and porous nanomaterials. Most of these structures are formed by aggregation of soft spherical supramolecules, which have soft coronas and overlap each other in the packing process. Our main focus here is to study the possible packing mechanisms via molecular dynamics simulations at the atomistic level. We consider the relative stability of various lattices packed by the soft dendrimer balls, first synthesized and characterized by Percec et al. (J. Am. Chem. Soc. 1997, 119, 1539) with different packing methods. The dendrons, which form the soft dendrimer balls, have the character of a hard aromatic region from the point of the cone to the edge with C(12) alkane "hair". After the dendrons pack into a sphere, the core of the sphere has the hard aromatic groups, while the surface is covered with the C(12) alkane "hair". In our studies, we propose three ways to organize the hair on the balls, Smooth/Valentino balls, Sticky/Einstein balls, and Asymmetric/Punk balls, which lead to three different packing mechanisms, Slippery, Sticky, and Anisotropic, respectively. We carry out a series of molecular dynamics (MD) studies on three plausible crystal structures (A15, FCC, and BCC) as a function of density and analyze the MD based on the vibrational density of state (DoS) method to extract the enthalpy, entropy, and free energies of these systems. We find that anisotropic packed A15 is favored over FCC, BCC lattices. Our predicted X-ray intensities of the best structures are in excellent agreement with experiment. "Anisotropic ball packing" proposed here plays an intermediate role between the enthalpy-favored "disk packing" and entropy-favored "isotropic ball packing", which explains the phase transitions at different temperatures. Free energies of various lattices at different densities are essentially the same, indicating that the preferred lattice is not determined during the packing process. Both enthalpy and entropy decrease as the density increases. Free energy change with volume shows two stable phases: the condensed phase and the isolated micelle phase. The interactions between the soft dendrimer balls are found to be lattice dependent when described by a two-body potential because the soft ball self-adjusts its shape and interaction in different lattices. The shape of the free energy potential is similar to that of the "square shoulder potential". A model explaining the packing efficiency of ideal soft balls in various lattices is proposed in terms of geometrical consideration.

Gaussian Fields and Random Packing

Consider sequential packing of unit volume balls in a large cube, in any dimension and with Poisson input. We show after suitable rescaling that the spatial distribution of packed balls tends to that of a Gaussian field in the thermodynamic limit. The results cover related applied models, including ballistic deposition and spatial birth-growth models.

Bestimmung der transitiven optimalen Kugelpackungen

Each of the 7 Coxeter space groups inBE3can be extended by the possible symmetries of its fundamental domain. Thus, we get a space groupGin the title. For any orbit type (Wyckoff position) ofGthe optimal ball packing and its D-V cell have been determined, described and enumerated in tables and figures. We make a complete list for all the orbit types of the 29 generalized Coxeter space groups above.

From deep holes to free planes

During the last decades, by applying techniques from Number Theory, Combinatorics and Measure Theory, remarkable progress has been made in the study of deep holes, free planes and related topics in packings of convex bodies, especially in lattice ball packings. Meanwhile, some fascinating new problems have been proposed. To stimulate further research in related areas, we will review the main results, some key techniques and some fundamental problems about deep holes, free cylinders and free planes in this paper.

Improving Rogers’ Upper Bound for the Density of Unit Ball Packings via Estimating the Surface Area of Voronoi Cells from Below in Euclidean \sl d -Space for All \sl d ≥ \bf 8

The sphere packing problem asks for the densest packing of unit balls in E d . This problem has its roots in geometry, number theory and information theory and it is part of Hilbert’s 18th problem. One of the most attractive results on the sphere packing problem was proved by Rogers in 1958. It can be phrased as follows. Take a regular d-dimensional simplex of edge length 2 in E d and then draw a d-dimensional unit ball around each vertex of the simplex. Let σ d denote the ratio of the volume of the portion of the simplex covered by balls to the volume of the simplex. Then the volume of any Voronoi cell in a packing of unit balls in E d is at least ω d /σ d , where ω d denotes the volume of a d-dimensional unit ball. This has the immediate corollary that the density of any unit ball packing in E d is at most σ d . In 1978 Kabatjanskii and Levenštein improved this bound for large d. In fact, Rogers’ bound is the presently known best bound for 4 ≤ d ≤ 42, and above that the Kabatjanskii—Levenštein bound takes over. In this paper we improve Rogers’ upper bound for the density of unit ball packings in Euclidean d-space for all d ≥ 8 and improve the Kabatjanskii—Levenštein upper bound in small dimensions. Namely, we show that the volume of any Voronoi cell in a packing of unit balls in E d , d ≥ 8, is at least $$\omega _d /\hat \sigma _d$$ and so the density of any unit ball packing in E d , d ≥ 8, is at most $$\hat \sigma _d$$ , where $$\hat \sigma _d$$ is a geometrically well-defined quantity satisfying the inequality $$\hat \sigma _d < \sigma _d$$ for all d≥ 8. We prove this by showing that the surface area of any Voronoi cell in a packing of unit balls in E d , d ≥ 8, is at least $$(d \cdot \omega _d )/\hat \sigma _d$$ .

On Free Planes in Lattice Ball Packings

This note, by studying the relations between the length of the shortest lattice vectors and the covering minima of a lattice, proves that for every d‐dimensional packing lattice of balls one can find a four‐dimensional plane, parallel to a lattice plane, such that the plane meets none of the balls of the packing, provided that the dimension d is large enough. Nevertheless, for certain ball packing lattices, the highest dimension of such ‘free planes’ is far from d.

The Voronoi–Delaunay approach for the free volume analysis of a packing of balls in a cylindrical container

The paper describes an approach for free volume analysis of packing of balls confined in a cylinder. The generalized Voronoi diagram is used as an underlying data structure. Two problems are addresses. One is an efficient construction of the confined Voronoi diagram inside a cylindrical boundary. The second is the analysis of the Voronoi network to study a distribution of empty spaces (voids) in the system. The approach is implemented in three-dimensional space and tested on the models of disordered packed bed of 300 balls in cylinders of different radii. The models were obtained by using the Monte Carlo method. The analysis of bottle-necks of channels between balls (and between balls and the boundary) and spatial distribution of the pores was performed.

Limit Theory for Random Sequential Packing and Deposition

Consider sequential packing of unit balls in a large cube, as in the Renyi car-parking model, but in any dimension and with finite input. We prove a law of large numbers and central limit theorem for the number of packed balls in the thermodynamic limit. We prove analogous results for numerous related applied models, including cooperative sequential adsorption, ballistic deposition, and spatial birth-growth models. The proofs are based on a general law of large numbers and central limit theorem for “stabilizing” functionals of marked point processes of independent uniform points in a large cube, which are of independent interest. “Stabilization” means, loosely, that local modifications have only local effects.

Damping capacity improvement of machine structures by close packing with balls

This paper deals with an experimental investigation on damping characteristics of structures packed with balls. The structures for testing are square pipes packed with glass balls. The effects of impulse magnitude, square pipe dimensions, ball size and excitation directions on damping characteristics are investigated. The experiments show that the damping characteristics are markedly affected by the ball size, and the damping capacity is improved by the ball packing. It is also shown that the damping capacity is affected by the packing arrangements, packing ratio and repulsion coefficient. Dramatic improvement in damping capacity is achieved by leaving only a small space between the packed balls and the inner surface of the square pipe. This fact is confirmed experimentally.

On the Optimum of Delsarte's Linear Program

We are interested in the maximal size A(n, d) of a binary error-correcting code of length n and distance d, or, alternatively, in the best packing of balls of radius (d−1)/2 in the n-dimensional Hamming space. The best known lower bound on A(n, d) is due to Gilbert and Varshamov and is obtained by a covering argument. The best know upper bound is due to McEliece, Rodemich, Rumsey, and Welch, and is obtained using Delsarte's linear programming approach. It is not known whether this is the best possible bound one can obtain from Delsarte's linear program. We show that the optimal upper bound obtainable from Delsarte's linear program will strictly exceed the Gilbert–Varshamov lower bound. In fact, it will be at least as big as the average of the Gilbert–Varshamov bound and the McEliece, Rodemich, Rumsey, and Welch upper bound. Similar results hold for constant weight binary codes. The average of the Gilbert–Varshamov bound and the McEliece, Rodemich, Rumsey, and Welch upper bound might be the true value of Delsarte's bound. We provide some evidence for this conjecture.

Edge Close Ball Packings

For a packing of unit balls in R 3, an edge-line is the intersection of the planes of two intersecting faces of some Dirichlet—Voronoi cell, like for example the line of an edge. Now the edge closeness of the arrangement is the maximal distance between an edge-line for a Dirichlet—Voronoi cell and the centre of the corresponding unit ball. We show that the unique packing with minimal edge closeness is generated by the space centred cubic lattice.

Segments in ball packings

Denote by Bn the n-dimensional unit ball centred at o. It is known that in every lattice packing of Bn there is a cylindrical hole of infinite length whenever n≥3. As a counterpart, this note mainly proves the following result: for any fixed ε with ε>0, there exist a periodic point set P(n, ε) and a constant c(n, ε) such that Bn + P(n, ε) is a packing in Rn, and the length of the longest segment contained in Rn\{int(εBn) + P(n, ε)} is bounded by c(n, ε) from above. Generalizations and applications are presented.

A lower bound for the mean width of Voronoi polyhedra of unit ball packings in E 3

In this note we prove that the mean width of any Voronoi cell of a packing of unit balls in E3 is at least \({{\pi }\over {6\sqrt {6}\cdot {\rm arcsin}({{1}\over {\sqrt 3}})-\sqrt {6}\pi }}= 2.3264\dots .\)As a comparison we mention that according to the Strong Dodecahedral Conjecture the mean width of any Voronoi cell of a packing of unit balls in E3 must be at least as large as 2.3736... the mean width of a regular dodecahedron of inradius 1.

On a stronger form of Roger's lemma and the minimum surface area of Voronoi cells in unit ball packings

On a stronger form of Roger's lemma and the minimum surface area of Voronoi cells in unit ball packings

粒子充てん構造物の振動減衰特性 １ ガラス粒充てんによる減衰能改善

This paper deals with an experimental investigation on damping characteristics of structures packed with balls. Test structures are square pipes packed with glass balls. The effects of impulse magnitude, square pipe dimensions and ball size on damping characteristics are investigated. The experiments show that the damping characteristics are markedly affected by the ball size and the damping capacity is improved by ball packing. It is also shown that the damping capacity is affected by packing arrangements, packing ratio and repulsion coefficient. The effective damping capacity improvement is achieved by remaining small space between packed balls and an inner surface of square pipe. This fact is confirmed by the experiment.

ON THE MINIMUM INTRINSIC 1-VOLUME OF VORONOI CELLS IN LATTICE UNIT SPHERE PACKINGS

A typical 3-dimensional (in short '3D') Voronoi cell of a 3Dlattice has six families of parallel edges. We call any six representants of these six families the generating edges of the Voronoi cell. The sum s of lengths of generating edges of a Voronoi-cell of a lattice unit sphere packing in the 3-dimensional Euclidean space is a special case of intrinsic 1-volumes of 3Dzonotopes with inradius 1 which are investigated accurately in [B]. However, the minimum of this value is unknown even in this special case. As the regular rhombic dodecahedron shows optimal properties in many similar problems, it was reasonable to conjecture that it also has the minimal s value. In this note we present a construction of a lattice unit ball packing whose Voronoi cell possesses an intrinsic 1-volume strictly less than the one of the proper regular rhombic dodecahedron, hence providing a smaller upper bound for s than it was conjectured. A further issue of the note is a formula for edge-lengths of Voronoi cells of lattice unit ball packings that can be used efficiently in similar calculations.

VORONOI POLYHEDRA OF UNIT BALL PACKINGS WITH SMALL SURFACE AREA

In this note, first, we give a very short new proof of the theorem which yields a lower bound for the surface area of Voronoi cells of unit ball packings in E d and implies Rogers' upper bound for the density of unit ball packings in E d for all d ≥ 2. Second we sharpen locally a classical result of Gauss by finding the locally smallest surface area Voronoi cells of lattice unit ball packings in E 3.

Statistics of shock waves in a two-dimensional granular flow.

We have investigated the dynamics of shock waves in a single layer of uniform balls in a small-angle two-dimensional funnel. When the funnel half-angle 0 degrees < or approximately beta < or approximately 2 degrees, the flow is intermittent and kinematic shock waves are observed to propagate against the flow. We have used fast video equipment and image analysis methods to study the statistics of the shock waves. It is found that their speed and frequency increase with the distance from the outlet. In particular, the shock speed scales as the ratio of the local funnel width to the width of the funnel outlet. Various kinds of interactions between shock waves are observed, including repulsion. New shock waves are only created at those sites where a close-packed triangular packing of the monodisperse balls fits across the funnel.