Spiral disk packings

2017 ◽  
Vol 345 ◽  
pp. 1-10 ◽  
Author(s):  
Yoshikazu Yamagishi ◽  
Takamichi Sushida
Keyword(s):  
2007 ◽  
Vol 22 (1) ◽  
pp. 65-72 ◽  
Author(s):  
Rong-Zhong Yu ◽  
Xin-Chu Fu ◽  
Shu-Liang Shui

10.37236/1223 ◽  
1994 ◽  
Vol 2 (1) ◽  
Author(s):  
R. L. Graham ◽  
B. D. Lubachevsky

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.


2020 ◽  
Vol 125 (2) ◽  
Author(s):  
Cristian Fernando Moukarzel ◽  
Gonzalo Peraza-Mues ◽  
Osvaldo Carvente

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

Author(s):  
Sen Qin ◽  
Jingyuan Zhang ◽  
Xufeng Chen ◽  
Fangyue Chen

2003 ◽  
Vol 17 (29) ◽  
pp. 5623-5630 ◽  
Author(s):  
TAMÁS UNGER ◽  
JÁNOS KERTÉSZ

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.


1967 ◽  
Vol 10 (3) ◽  
pp. 395-415 ◽  
Author(s):  
John B. Wilker

It is an old problem to find how a collection of congruent plane figures should be arranged without overlapping to cover the largest possible fraction of the plane or some region of the plane. If similar figures of arbitrary different sizes are permitted, Vitali's theorem ([7] p. 109) guarantees that packings which cover almost all points are possible. It is natural to study the diameters of figures used in such a packing and we will investigate this for the case of a closed disk packed with smaller open disks.


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