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May 23, 2006, 6:02:10 PM5/23/06

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During the last few month I received several requests asking for a

formula to estimate how many spheres of equal size can be packed into a

spherical container. Reasonably good packing results for small sphere

numbers can be seen on the web page (Java needed):

formula to estimate how many spheres of equal size can be packed into a

spherical container. Reasonably good packing results for small sphere

numbers can be seen on the web page (Java needed):

http://www.randomwalk.de/sphere/insphr/spheresinsphr.html

The currently best known radii for n<=72 are given in the table

http://www.randomwalk.de/sphere/insphr/spisbest.txt

Numerical results for larger n are currently not available.

My current recommendation for an heuristic estimate of the number of

equal spheres of radius r that fit into a container of radius R is as

follows:

Calculate relative diameter of small spheres d=r/R.

Relative diameter of core region:

q = 1 - 2*d

Upper limit for number of spheres:

n = (1/d^3) * ( 0.7405*q^3 + 0.484*(1 - q^3) )

Some examples:

d n Best known d

for this n

0.25 33 0.2487624

0.24 38 0.2405156

0.23 43 0.229733

0.22 50 0.2197529

0.21 58 0.2101144

0.2 67 0.2004418

0.19 79

0.18 95

0.17 114

0.16 138

0.15 169

0.14 211

0.13 268

0.12 345

0.11 455

0.1 615

0.09 858

0.08 1242

0.07 1887

0.06 3050

0.05 5368

0.04 10683

0.03 25816

0.02 88867

0.01 725416

d n

0.176863 100

0.0857216 1000

0.0408686 10000

0.01923815 100000

0.00899139 1000000

This assumes an optimally dense packing of the core region and a

heuristically determined lower density for a surface layer. For randomly

packed spheres a more realistic estimate might be obtained by replacing

the Kepler packing density 0.7405 for the core by something like the

value 0.64 found by Jaeger & Nagel.

A comparison with the numerical packing results is shown in

http://www.randomwalk.de/sphere/insphr/nsphisph.gif

For n=60,61 the actually achievable packing density is significantly

higher than predicted by the formula given above.

Hugo Pfoertner

May 29, 2006, 1:31:58 AM5/29/06

to

Hugo Pfoertner <not...@abouthugo.de> wrote:

> During the last few month I received several requests asking for a

> formula to estimate how many spheres of equal size can be packed into a

> spherical container. Reasonably good packing results for small sphere

> numbers can be seen on the web page (Java needed):

>

> http://www.randomwalk.de/sphere/insphr/spheresinsphr.html

>

> The currently best known radii for n<=72 are given in the table

>

> http://www.randomwalk.de/sphere/insphr/spisbest.txt

> During the last few month I received several requests asking for a

> formula to estimate how many spheres of equal size can be packed into a

> spherical container. Reasonably good packing results for small sphere

> numbers can be seen on the web page (Java needed):

>

> http://www.randomwalk.de/sphere/insphr/spheresinsphr.html

>

> The currently best known radii for n<=72 are given in the table

>

> http://www.randomwalk.de/sphere/insphr/spisbest.txt

Hi Hugo,

Please check the radius which you list for Thierry Gensane's packing of 28

spheres. I suspect that it should actually be 0.2603548 -- in other words,

that perhaps someone transposed a pair of digits. But that's just a guess.

(In the highly unlikely event that 0.2630548 is actually correct, I claim

an improved packing of 27 spheres! ;-)

> Numerical results for larger n are currently not available.

> My current recommendation for an heuristic estimate of the number of

> equal spheres of radius r that fit into a container of radius R is as

> follows:

>

> Calculate relative diameter of small spheres d=r/R.

> Relative diameter of core region:

>

> q = 1 - 2*d

>

> Upper limit for number of spheres:

>

> n = (1/d^3) * ( 0.7405*q^3 + 0.484*(1 - q^3) )

For those who don't recognize it right away, that coefficient of q^3 is

precisely K = pi/(3*sqrt(2)). Based on thinking about asymptotics, I agree

with Hugo that that Kepler packing density must be the correct coefficient

there.

I have devised a simpler formula, closely related to Hugo's, which I

suspect may always give a lower bound for the number of spheres:

n = K*(1 - 2*d)/d^3 + 1/(2*d^2)

In other words, if someone asks "Given d, how many spheres can I pack?",

perhaps a correct answer is "At least floor(K*(1 - 2*d)/d^3 + 1/(2*d^2))

spheres."

> Some examples:

Below, following Hugo's n, I give some n, shown in square brackets, as

produced by my simple conjectured lower bound. Notice however that, for d

less than roughly 0.21, my formula gives n _larger_ than Hugo's. If my

formula does give a lower bound, as conjectured, then some of the best

packings currently known would necessarily be suboptimal. (In particular,

they are the packings for n = 55, 56, 62, 64, 65, 67, 68, 69, 71 and 72

spheres.)

> d n Best known d

> for this n

> 0.25 33 [31] 0.2487624

> 0.24 38 [36] 0.2405156

> 0.23 43 [42] 0.229733

> 0.22 50 [49] 0.2197529

> 0.21 58 [57] 0.2101144

> 0.2 67 [68] 0.2004418

> 0.19 79

> 0.18 95

> 0.17 114

> 0.16 138

> 0.15 169 [175]

> 0.14 211

> 0.13 268

> 0.12 345

> 0.11 455

> 0.1 615 [642]

> 0.09 858

> 0.08 1242

> 0.07 1887

> 0.06 3050

> 0.05 5368

> 0.04 10683

> 0.03 25816

> 0.02 88867

> 0.01 725416 [730670]

Regards,

David W. Cantrell

May 29, 2006, 3:01:18 PM5/29/06

to

"David W. Cantrell" wrote:

>

> Hugo Pfoertner <not...@abouthugo.de> wrote:

> > During the last few month I received several requests asking for a

> > formula to estimate how many spheres of equal size can be packed into a

> > spherical container. Reasonably good packing results for small sphere

> > numbers can be seen on the web page (Java needed):

> >

> > http://www.randomwalk.de/sphere/insphr/spheresinsphr.html

> >

> > The currently best known radii for n<=72 are given in the table

> >

> > http://www.randomwalk.de/sphere/insphr/spisbest.txt

>

> Hi Hugo,

>

> Please check the radius which you list for Thierry Gensane's packing of 28

> spheres. I suspect that it should actually be 0.2603548 -- in other words,

> that perhaps someone transposed a pair of digits. But that's just a guess.

> (In the highly unlikely event that 0.2630548 is actually correct, I claim

> an improved packing of 27 spheres! ;-)

>

> Hugo Pfoertner <not...@abouthugo.de> wrote:

> > During the last few month I received several requests asking for a

> > formula to estimate how many spheres of equal size can be packed into a

> > spherical container. Reasonably good packing results for small sphere

> > numbers can be seen on the web page (Java needed):

> >

> > http://www.randomwalk.de/sphere/insphr/spheresinsphr.html

> >

> > The currently best known radii for n<=72 are given in the table

> >

> > http://www.randomwalk.de/sphere/insphr/spisbest.txt

>

> Hi Hugo,

>

> Please check the radius which you list for Thierry Gensane's packing of 28

> spheres. I suspect that it should actually be 0.2603548 -- in other words,

> that perhaps someone transposed a pair of digits. But that's just a guess.

> (In the highly unlikely event that 0.2630548 is actually correct, I claim

> an improved packing of 27 spheres! ;-)

Actually the radius in Thierry Gensane's list for n=28 is

0.2603054756814.

As one can easily see from a comparison between the results found by

Thierry Gensane and my results for n<=50 that many of my results are not

optimal. There is no reason that this should be different beyond n=50.

This gives me some motivation to continue searching for improved

packings, at least after the end of the current round of Al Zimmerman's

programming contest ;-)

BTW, it would be interesting to see a comparison of the estimates from

David's formula against the number of sphere centers inside the Waterman

polyhedra, for which I created some OEIS entries just recently, see

http://www.research.att.com/~njas/sequences/A119869 , the phantastic

Java applet written by Mark Newbold

http://dogfeathers.com/java/ccppoly.html and Paul Bourke's

http://astronomy.swin.edu.au/~pbourke/geometry/waterman/index3.html

(bottom of page).

Since the Waterman polyhedra have an optimally dense packed core from

their construction, the number of spheres inside should be not too far

from the value given by the formula.

Hugo

Feb 3, 2018, 1:14:55 PM2/3/18

to

If i know radius (r) Container (sphere) and radius (r1) of spheres. How Can i find maximum quantity of spheres in Container.

Feb 4, 2018, 10:21:24 AM2/4/18

to

On Sat, 3 Feb 2018 10:14:46 -0800 (PST), kiril...@gmail.com wrote:

> If i know radius (r) Container (sphere) and radius (r1) of spheres.

> How Can i find maximum quantity of spheres in Container.

⌊2 / (1 + 2⋅q − q²)⌋ where q = r/r1
> If i know radius (r) Container (sphere) and radius (r1) of spheres.

> How Can i find maximum quantity of spheres in Container.

Solution derived from:

<en.wikipedia.org/wiki/Packing_problems#Spheres_into_a_Euclidean_ball>

Gerd

Feb 4, 2018, 6:04:28 PM2/4/18

to

inner spheres only holds if k <= n + 1 where n is the dimension of the

Euclidean space. So for n = 3 it only holds for k <= 4. For more

information look at:

https://en.wikipedia.org/wiki/Sphere_packing_in_a_sphere

Bob

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