Dense packings of 3k(k+1)+1 equal disks in a circle for k = 1, 2, 3, 4, and 5

1995 ◽  
pp. 303-312 ◽  
Author(s):  
B. D. Lubachevsky ◽  
R. L. Graham
Keyword(s):  
Dense Packings

Dense Packings of Random Binary Assemblies of Disks

1997 ◽  
Vol 7 (2) ◽  
pp. 247-254 ◽  
Author(s):  
N. N. Sinelnikov ◽  
M. A. Mazo ◽  
Al. Al. Berlin
Keyword(s):  
Dense Packings

scholarly journals Theory of cylindrical dense packings of disks

2014 ◽  
Vol 89 (4) ◽  
Author(s):  
A. Mughal ◽  
D. Weaire
Keyword(s):  
Dense Packings

scholarly journals Generating dense packings of hard spheres by soft interaction design

2018 ◽  
Vol 4 (6) ◽  
Author(s):  
Thibaud Maimbourg ◽  
Mauro Sellitto ◽  
Guilhem Semerjian ◽  
Francesco Zamponi
Keyword(s):  
Interaction Design ◽  
Optimization Problem ◽  
Hard Spheres ◽  
Hard Core ◽  
Packing Fraction ◽  
Exact Formulation ◽  
Simple Liquids ◽  
Soft Interaction ◽  
Dense Packings ◽  
Number Of Particles

Packing spheres efficiently in large dimension dd is a particularly difficult optimization problem. In this paper we add an isotropic interaction potential to the pure hard-core repulsion, and show that one can tune it in order to maximize a lower bound on the packing density. Our results suggest that exponentially many (in the number of particles) distinct disordered sphere packings can be efficiently constructed by this method, up to a packing fraction close to 7 \, d \, 2^{-d}7d2−d. The latter is determined by solving the inverse problem of maximizing the dynamical glass transition over the space of the interaction potentials. Our method crucially exploits a recent exact formulation of the thermodynamics and the dynamics of simple liquids in infinite dimension.

scholarly journals Dense Packings of Equal Disks in an Equilateral Triangle

10.37236/1223 ◽  
1994 ◽  
Vol 2 (1) ◽  
Author(s):  
R. L. Graham ◽  
B. D. Lubachevsky
Keyword(s):  
Equilateral Triangle ◽  
Minimum Distance ◽  
Discrete Event ◽  
Simulation Algorithm ◽  
Infinite Class ◽  
Event Simulation ◽  
Integer Coefficients ◽  
Equilateral Triangles ◽  
Dense Packings ◽  
Disk Packings

Previously published packings of equal disks in an equilateral triangle have dealt with up to 21 disks. We use a new discrete-event simulation algorithm to produce packings for up to 34 disks. For each $n$ in the range $22 \le n \le 34$ we present what we believe to be the densest possible packing of $n$ equal disks in an equilateral triangle. For these $n$ we also list the second, often the third and sometimes the fourth best packings among those that we found. In each case, the structure of the packing implies that the minimum distance $d(n)$ between disk centers is the root of polynomial $P_n$ with integer coefficients. In most cases we do not explicitly compute $P_n$ but in all cases we do compute and report $d(n)$ to 15 significant decimal digits. Disk packings in equilateral triangles differ from those in squares or circles in that for triangles there are an infinite number of values of $n$ for which the exact value of $d(n)$ is known, namely, when $n$ is of the form $\Delta (k) := \frac{k(k+1)}{2}$. It has also been conjectured that $d(n-1) = d(n)$ in this case. Based on our computations, we present conjectured optimal packings for seven other infinite classes of $n$, namely \begin{align*} n & = & \Delta (2k) +1, \Delta (2k+1) +1, \Delta (k+2) -2 , \Delta (2k+3) -3, \\ && \Delta (3k+1)+2 , 4 \Delta (k), \text{ and } 2 \Delta (k+1) + 2 \Delta (k) -1 . \end{align*} We also report the best packings we found for other values of $n$ in these forms which are larger than 34, namely, $n=37$, 40, 42, 43, 46, 49, 56, 57, 60, 63, 67, 71, 79, 84, 92, 93, 106, 112, 121, and 254, and also for $n=58$, 95, 108, 175, 255, 256, 258, and 260. We say that an infinite class of packings of $n$ disks, $n=n(1), n(2),...n(k),...$, is tight , if [$1/d(n(k)+1) - 1/d(n(k))$] is bounded away from zero as $k$ goes to infinity. We conjecture that some of our infinite classes are tight, others are not tight, and that there are infinitely many tight classes.

scholarly journals Dense packings

2006 ◽  
Vol 36 (3a) ◽  
Author(s):  
H. J. Herrmann ◽  
R. Mahmoodi Baram ◽  
M. Wackenhut
Keyword(s):  
Dense Packings

scholarly journals Electron microscopy of nanoparticle superlattice formation at a solid-liquid interface in nonpolar liquids

2020 ◽  
Vol 6 (20) ◽  
pp. eaba1404
Author(s):  
E. Cepeda-Perez ◽  
D. Doblas ◽  
T. Kraus ◽  
N. de Jonge
Keyword(s):  
Electron Microscopy ◽  
Three Dimensional ◽  
Honeycomb Structure ◽  
Liquid Interface ◽  
Thin Layers ◽  
Fine Tuning ◽  
Particle Layer ◽  
Dense Packings ◽  
Solid Liquid Interface ◽  
Solid Liquid

Nanoparticle superlattice films form at the solid-liquid interface and are important for mesoscale materials, but are notoriously difficult to analyze before they are fully dried. Here, the early stages of nanoparticle assembly were studied at solid-liquid interfaces using liquid-phase electron microscopy. Oleylamine-stabilized gold nanoparticles spontaneously formed thin layers on a silicon nitride (SiN) membrane window of the liquid enclosure. Dense packings of hexagonal symmetry were obtained for the first monolayer independent of the nonpolar solvent type. The second layer, however, exhibited geometries ranging from dense packing in a hexagonal honeycomb structure to quasi-crystalline particle arrangements depending on the dielectric constant of the liquid. The complex structures formed by the weaker interactions in the second particle layer were preserved, while the surface remained immersed in liquid. Fine-tuning the properties of the involved materials can thus be used to control the three-dimensional geometry of a superlattice including quasi-crystals.

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