Bounds for circles in circle packing

carlosinho · #1 $Old$ November 3rd, 2007, 10:30 AM

We would like to pack as many circles of diameter

d as possible into another large circle of
diameter D. We denote by N(d;D) such number.
(a) Give the upper bound for N(d;D).
(b) Give in details how we can calculate a lower bound for N(d;D). Find an explicit value for

N

when d = 9 mm and D = 72 mm.

ANY IDEAS?!

Plato · #2 $Old$ November 3rd, 2007, 10:49 AM

With a bit of effort you too could have found this site.
Circle Packing -- from Wolfram MathWorld

carlosinho · #3 $Old$ November 3rd, 2007, 11:50 AM

But, i need equations which gave values on Circle Packing -- from Wolfram MathWorld beacuse part a) deals with variables

CaptainBlack · #4 $Old$ November 3rd, 2007, 12:42 PM

Quote:

Originally Posted by carlosinho $View Post$

We would like to pack as many circles of diameter

d as possible into another large circle of

diameter D. We denote by N(d;D) such number.
(a) Give the upper bound for N(d;D).
(b) Give in details how we can calculate a lower bound for N(d;D). Find an explicit value for

N

when d = 9 mm and D = 72 mm.

ANY IDEAS?!

Parts a) and b) are so vaguely worded that we can give bounds as follows:

Clearly if d<D: 1<floor(D/d)<N(d;D)< floor(D^2/d^2)

but these are not tight.

RonL

pheinrich · #5 $Old$ May 19th, 2009, 05:34 PM

To my knowledge, there is no formula that will compute N(d;D) precisely. However, Eckard Specht has compiled a table of observed packings for N up to 900 (and beyond). From his data, it's clear that D/d varies smoothly as a power function of N.

There's a javascript calculator at the bottom of Fiber Stuffing Revisited | Saphum that uses Specht's data to estimate N(d;D).

For D = 72mm and d = 9mm, N = 51.

-peter