scholarly journals The C∞-convergence of hexagonal disk packings to the Riemann map

1998 ◽  
Vol 180 (2) ◽  
pp. 219-245 ◽  
Author(s):  
Zheng-Xu He ◽  
Oded Schramm
Keyword(s):  
Riemann Map ◽  
Disk Packings

Density of invariant disk packings for planar piecewise isometries

2007 ◽  
Vol 22 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Rong-Zhong Yu ◽  
Xin-Chu Fu ◽  
Shu-Liang Shui
Keyword(s):  
Piecewise Isometries ◽  
Disk Packings

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 Rotational Transport via Spontaneous Symmetry Breaking in Vibrated Disk Packings

2020 ◽  
Vol 125 (2) ◽  
Author(s):  
Cristian Fernando Moukarzel ◽  
Gonzalo Peraza-Mues ◽  
Osvaldo Carvente
Keyword(s):  
Symmetry Breaking ◽  
Spontaneous Symmetry Breaking ◽  
Disk Packings

scholarly journals Vibrations of jammed disk packings with Hertzian interactions

2013 ◽  
Vol 16 (2) ◽  
pp. 209-216 ◽  
Author(s):  
Carl F. Schreck ◽  
Corey S. O’Hern ◽  
Mark D. Shattuck
Keyword(s):  
Disk Packings

Enumeration of spanning trees on contact graphs of disk packings

2015 ◽  
Vol 433 ◽  
pp. 1-8 ◽  
Author(s):  
Sen Qin ◽  
Jingyuan Zhang ◽  
Xufeng Chen ◽  
Fangyue Chen
Keyword(s):  
Spanning Trees ◽  
Contact Graphs ◽  
Disk Packings

FRICTIONAL INDETERMINANCY OF FORCES IN HARD-DISK PACKINGS

2003 ◽  
Vol 17 (29) ◽  
pp. 5623-5630 ◽  
Author(s):  
TAMÁS UNGER ◽  
JÁNOS KERTÉSZ
Keyword(s):  
Random Walk ◽  
Friction Coefficient ◽  
Contact Forces ◽  
Contact Dynamics ◽  
Force Fluctuations ◽  
Packing Structure ◽  
Hard Particles ◽  
Rigid Disks ◽  
Dynamics Simulations ◽  
Disk Packings

We study the statical indeterminacy of contact forces in 2D random frictional packings of perfectly rigid disks. Based on contact dynamics simulations we perform a random walk in the force space in order to explore the equilibrium force-states for a fixed packing structure. Our measurement is in agreement with the isostaticity of frictionless hard particles, in that case forces are fully determined. For non-zero friction coefficient the problem gets undetermined, the possible force fluctuations are growing with increasing friction up to a maximum at friction coefficient around 0.1. Further increase of friction reduces the force fluctuations on the average.

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