Ball Packings Research Articles

The efficacy of cotton ball packing after endoscopic sinus surgery: A prospective, randomized, controlled trial

ObjectiveThe blocking of airflow into sinonasal cavity may decrease postoperative crusting and the development of adhesions. The purpose of this study was to investigate the efficacy of cotton ball packing in patients following endoscopic sinus surgery (ESS). MethodsThirty nine patients with chronic rhinosinusitis requiring the same extent of ESS were included. As a part of postoperative care, the patients were instructed to perform a nasal saline irrigation and apply a nasal spray in each nostril, and then informed to put a cotton ball in a one side of nostril, and the other side was kept to be empty as a control. Patients’ subjective symptoms, patients’ pain while receiving sinonasal cavity debridement, time required to perform debridement, and postoperative wound healing were evaluated. ResultsAlthough cotton ball packing resulted in less discomfort for postnasal drip, rhinorrhea, headache, and facial pain than no packing, there were no statistically significant differences between the groups. The cotton ball packing was associated with significantly less pain on while performing postoperative debridement, therefore less time was needed to perform debridement. The cotton ball packing appears to improve wound healing within the sinus cavities up to 1 month postoperatively. ConclusionThe use of the cotton ball packing after ESS results in significantly less formation of crusts and adhesions, leading to decreasing pain and time during postoperative debridement and promoting faster wound healing.

Decomposition method related to saturated hyperball packings

In this paper we study the problem of hyperball (hypersphere) packings in 3-dimensional hyperbolic space. We introduce a new definition of the non-compact saturated ball packings with generalized balls (horoballs, hyperballs) and describe to each saturated hyperball packing, a new procedure to get a decomposition of 3-dimensional hyperbolic space ℍ 3 into truncated tetrahedra. Therefore, in order to get a density upper bound for hyperball packings, it is sufficient to determine the density upper bound of hyperball packings in truncated simplices.

Kottman’s constant, packing constant and Riesz angle in some classes of K ¨othe sequence spaces

We have found a sufficient condition in order that the Kottman constant to be equal to the Riesz angle for Kothe ¨ sequence spaces. We have found the ball packing constant in weighted Orlicz sequence spaces, endowed with Luxemburg or p–Amemiya norm. We have calculated the Riesz angle for Musielak–Orlicz, Nakano, weighted Orlicz, Orlicz, Orlicz–Lorentz, Lorentz and Cesaro sequence spaces.

Top dense hyperbolic ball packings and coverings for complete coxeter orthoscheme groups

In n-dimensional hyperbolic space Hn (n > 2), there are three types of spheres (balls): the sphere, horosphere and hypersphere. If n = 2, 3 we know a universal upper bound of the ball packing densities, where each ball?s volume is related to the volume of the corresponding Dirichlet-Voronoi (D-V) cell. E.g., in H3 a densest (not unique) horoball packing is derived from the {3,3,6} Coxeter tiling consisting of ideal regular simplices T? reg with dihedral angles ?/3. The density of this packing is ??3 ? 0.85328 and this provides a very rough upper bound for the ball packing densities as well. However, there are no ?essential" results regarding the ?classical" ball packings with congruent balls, and for ball coverings either. The goal of this paper is to find the extremal ball arrangements in H3 with ?classical balls". We consider only periodic congruent ball arrangements (for simplicity) related to the generalized, so-called complete Coxeter orthoschemes and their extended groups. In Theorems 1.1 and 1.2 we formulate also conjectures for the densest ball packing with density 0.77147... and the loosest ball covering with density 1.36893..., respectively. Both are related with the extended Coxeter group (5,3,5) and the so-called hyperbolic football manifold. These facts can have important relations with fullerenes in crystallography.

The sphere packing problem in dimension $8$

In this paper we prove that no packing of unit balls in Euclidean space $\mathbb{R}^8$ has density greater than that of the $E_8$-lattice packing.

Ball packings with high chromatic numbers from strongly regular graphs

Inspired by Bondarenko’s counter-example to Borsuk’s conjecture, we notice some strongly regular graphs that provide examples of ball packings whose chromatic numbers are significantly higher than the dimensions. In particular, from generalized quadrangles we obtain unit ball packings in dimension q3−q2+q with chromatic number q3+1, where q is a prime power. This improves the previous lower bounds for the chromatic number of ball packings.

Minkowski bisectors, Minkowski cells and lattice coverings

The Dirichlet–Voronoi cell and parallelohedron are fundamental concepts in Geometry. In particular, they do play important roles in the study of ball packing and ball covering. However, to study packing and covering of general convex bodies, they are no longer so useful (see Theorem 0). By introducing Minkowski bisectors and Minkowski cells, this paper explores a new way to study the density $$\theta ^*(C)$$ of the thinnest lattice covering of $$\mathbb {E}^n$$ by a centrally symmetric convex body C. Several basic results (Theorems 2 and 4, Corollary 1) and unexpected geometric phenomena (Theorem 0, Example 1, Remark 4) about Minkowski bisectors, Minkowski cells and covering densities are discovered.

Weak waves under periodic load applied to a packing of glass balls

Based on the tests of packings of calibrated glass balls with a diameter of 1 mm, it is shown that multiple point impacts improve waveguide characteristics of the medium—conducting paths composed of force “chains” emerge in the test material. The further quasi-static alternating shears change the packing of particles, break the chains and reduce conduction. If continued onward, the multiple impulsive loading results in recovery of the “chains” and in better conduction of the paths.

Even More Infinite Ball Packings from Lorentzian Root Systems

Boyd (1974) proposed a class of infinite ball packings that are generated by inversions. Later, Maxwell (1983) interpreted Boyd's construction in terms of root systems in Lorentz spaces. In particular, he showed that the space-like weight vectors correspond to a ball packing if and only if the associated Coxeter graph is of "level 2"'. In Maxwell's work, the simple roots form a basis of the representations space of the Coxeter group. In several recent studies, the more general based root system is considered, where the simple roots are only required to be positively independent. In this paper, we propose a geometric version of "level'' for root systems to replace Maxwell's graph theoretical "level''. Then we show that Maxwell's results naturally extend to the more general root systems with positively independent simple roots. In particular, the space-like extreme rays of the Tits cone correspond to a ball packing if and only if the root system is of level $2$. We also present a partial classification of level-$2$ root systems, namely the Coxeter $d$-polytopes of level-$2$ with $d+2$ facets.

Non-periodic geodesic ball packings generated by infinite regular prism tilings in $\SLR $ space

Non-periodic geodesic ball packings generated by infinite regular prism tilings in $\SLR $ space

Apollonian Ball Packings and Stacked Polytopes

We investigate in this paper the relation between Apollonian $d$-ball packings and stacked $(d+1)$-polytopes for dimension $d\ge 3$. For $d=3$, the relation is fully described: we prove that the $1$-skeleton of a stacked $4$-polytope is the tangency graph of an Apollonian $3$-ball packing if and only if no six $4$-cliques share a $3$-clique. For higher dimension, we have some partial results.

Geodesic and Translation Ball Packings Generated by Prismatic Tessellations of the Universal Cover of $${{\rm {SL}}_{2}({\mathbb{R}})}$$ SL 2 ( R )

We construct ball packings of the universal cover of $${{\rm{SL}}_{2}({\mathbb{R}})}$$ by geodesic balls and translation balls. The packings are generated by action of the prism groups $${{\mathbf{pq}}_{k}{\mathbf{o}}_{\ell}}$$ . We obtain volume formulae for calculations in geographical coordinates. Using these formulae we find numerically the maximal dense packings for cases $${k=1}$$ , $${o=2}$$ , $${\ell=1}$$ and small values of $${p}$$ and $${q}$$ .

$${\varepsilon}$$ ε -quasi-twelve-neighbour packings of unit balls in $${\mathbb{E}^3}$$ E 3

Let \({\varepsilon}\) be a (small) positive number. A packing of unit balls in \({{\mathbb{E}^{3}}}\) is said to be an \({\varepsilon}\)-quasi-twelve-neighbour packing if no two balls of the packing touch each other but for each unit ball B of the packing there are twelve other balls in the packing with the property that the distance of the centre of each of these twelve balls from the centre of B is smaller than \({2+\varepsilon}\). We construct \({\varepsilon}\)-quasi-twelve-neighbour packings of unit balls in \({{\mathbb{E}^{3}}}\) for arbitrary small positive \({\varepsilon}\) with some surprising properties.

EFFECT OF PACKING MATERIAL AND ITS GEOMETRY ON THE PERFORMANCE OF PACKED BED COLLECTOR

It has been well established that packed bed solar collectors perform better as compared to conventional collectors. Results of performance studies on packed bed solar collector are available in literature in which different operating conditions have been considered which make it difficult to compare their performance accurately. Considering this comparative study of performance of solar collector with different packing elements has been made in the present work. Experimental investigations on solar collector packed with iron chips, wire mesh, gravels and glass balls for the same set of operating parameter have been done on a single setup to study the effect of packing material and its geometry on the thermal efficiency of packed bed collector. It is observed that iron chips packed collector is identified as the best packing materials out of the materials selected for study leading to thermal efficiency of 76.21% for the mass flow rate of 0.035 kg/s and porosity of 0.945, which is 69.58% higher as compared to smooth collector. Thermal efficiency of wire mesh packed collector for similar operating conditions is found to be 74.26% which is 65.24% higher than smooth collector. In low porosity range gravel packed collector is found to perform better as compared to glass ball packing. Effect of mass flow rate on the effective efficiency has also been conducted for various packing elements used in the present study. Based on the experimental results, plots have been drawn for efficiency against temperature rise parameters for different packing elements which can be used by the designer for choosing the correct value of mass flow rate for the specific temperature rise application.

12-Neighbour Packings of Unit Balls in $${\mathbb{E}^3}$$ E 3

We prove that any 12-neighbour packing of unit balls in \({\mathbb{E}^3}\) is composed of parallel layers of the same hexagonal structure as the layers in the densest lattice packing.

Application of an idea of Voronoĭ to lattice packing, supplement

Voronoĭ’s criterion on lattice packings of balls in \(\mathbb E^d\) of local maximum density was extended in an earlier article to lattice packings of convex bodies in \(\mathbb E^d\) with local ultra maximum density, a stronger version of maximum density. No example of such a packing was given. However, it was conjectured that for generic convex bodies, the lattice packings of local maximum density are of local ultra maximum density provided that the dimension \(d\) is sufficiently high. For generic convex bodies, the kissing number of lattice packings of global maximum density is between \(d(d+1)\) and \(2d^2\). We conjecture that it equals \(2d^2\). In this note, first steps toward these conjectures are made. The existence of convex bodies with lattice packings of local ultra maximum density is shown. Contrasting this, families of convex bodies are specified which do not admit lattice packings of local ultra maximum density. If for a dense family of convex bodies every lattice packing of global maximum density has ultra maximum density, then this holds for a dense open family of convex bodies as well. A related result treats kissing numbers.

Densest Geodesic Ball Packings to S 2 × R space groups generated by screw motions

In this paper we study the locally optimal geodesic ball packings with equal balls to the S 2 × R space groups having rotation point groups and their generators are screw motions. We determine and visualize the densest simply transitive geodesic ball arrangements for the above space groups; moreover, we compute their optimal densities and radii. The densest packing is derived from the S 2 × R space group 3qe. I. 3 with packing density ≈0.7278. E. Molnar has shown in [9] that the Thurston geometries have an unified interpretation in the real projective 3-sphere \({\mathcal{PS}^3}\). In our work we shall use this projective model of S 2 × R geometry.

Density bounds for outer parallel domains of unit ball packings

We give upper bounds for the density of unit ball packings relative to their outer parallel domains and discuss their connection to contact numbers. Also, packings of soft balls are introduced and upper bounds are given for the fraction of space covered by them.

Density Bounds for Outer Parallel Domains of Unit Ball Packings

Density Bounds for Outer Parallel Domains of Unit Ball Packings

Ball Packings with Periodic Constraints

We call a periodic ball packing in $$d$$d-dimensional Euclidean space periodically (resp. strictly) jammed with respect to a period lattice $$\varLambda $$? if there are no nontrivial motions of the balls that preserve $$\varLambda $$? (resp. that maintain some period with smaller or equal volume). In particular, we call a packing consistently periodically jammed (resp. consistently strictly jammed) if it is periodically (resp. strictly) jammed on every one of its periods. After extending a well-known bar framework and stress condition to strict jamming, we prove that a packing with period $$\varLambda $$? is consistently strictly jammed if and only if it is strictly jammed with respect to $$\varLambda $$? and consistently periodically jammed. We next extend a result about rigid unit mode spectra in crystallography to characterize periodic jamming on sublattices. After that, we prove that there are finitely many strictly jammed packings of $$m$$m unit balls and other similar results. An interesting example shows that the size of the first sublattice on which a packing is first periodically unjammed is not bounded. Finally, we find an example of a consistently periodically jammed packing of low density $$\delta = \frac{4 \pi }{6 \sqrt{3} + 11} + \varepsilon \approx 0.59$$?=4?63+11+??0.59, where $$\varepsilon $$? is an arbitrarily small positive number. Throughout the paper, the statements for the closely related notions of periodic infinitesimal rigidity and affine infinitesimal rigidity for tensegrity frameworks are also given.