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
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
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