Maybe not even optimal results, just approximations.

[José Rui Faustino de Sousa]

Regular pyramidal packing of hard spheres



Hi!



I am trying to find the best way to pack hard spheres in regular
pyramids.



I am only interested in the triangular, square, pentagonal, hexagonal,
octagonal and decagonal cases.



The triangular, square and hexagonal cases are easy corresponding to the
face centred cubic, body centred cubic and hexagonal close packing
crystal lattice packing schemes.



The pentagonal most likely corresponds to a stacking of similar to the
one generated by the pentagonal pyramidal number.



The octagonal and decagonal pyramidal packing would be more or less
similar to the hexagonal packing alternating centred octagons (decagons)
with squares (pentagons).



Is there any literature on the subject, maybe not even optimal results
just approximations? I have not had any luck searching.



Thank you very much.



Best regards

José Rui

[Arnold Neumaier]

José Rui Faustino de Sousa wrote:
>
> I am trying to find the best way to pack hard spheres in regular
> pyramids.
>
> I am only interested in the triangular, square, pentagonal, hexagonal,
> octagonal and decagonal cases.
>
> The triangular, square and hexagonal cases are easy corresponding to the
> face centred cubic, body centred cubic and hexagonal close packing
> crystal lattice packing schemes.

The body-centered cubic lattice is not an optimal packing in a large
enough box (quadratic pyramid), in view of Hales' proof of Kepler's
conjecture. For all sufficiently thick bodies, near FCC and HCP
packings will be best, since the packing density in the interior is
the only thing which matters asymptotically.


Arnold Neumaier

[frisbieinstein@yahoo.com]

José Rui Faustino de Sousa wrote:
> Regular pyramidal packing of hard spheres
>
>
>
> Hi!
>
>
>
> I am trying to find the best way to pack hard spheres in regular
> pyramids.
>
>
>
> I am only interested in the triangular, square, pentagonal, hexagonal,
> octagonal and decagonal cases.
>
>
>
> The triangular, square and hexagonal cases are easy corresponding to the
> face centred cubic, body centred cubic and hexagonal close packing
> crystal lattice packing schemes.
>
>
>
> The pentagonal most likely corresponds to a stacking of similar to the
> one generated by the pentagonal pyramidal number.
>
>
>
> The octagonal and decagonal pyramidal packing would be more or less
> similar to the hexagonal packing alternating centred octagons (decagons)
> with squares (pentagons).
>
>
>
> Is there any literature on the subject, maybe not even optimal results
> just approximations? I have not had any luck searching.
>
>
>
> Thank you very much.
>
>
>
> Best regards
>
> José Rui

Sphere packing is known to be a very hard problem. Isaac Newton
considered the question of spheres in an infinite space (I think) and
concluded that the ordinary packing is optimal. This was proved only
thirty years or so ago and the ostensible proof is a calculation so
complex that many don't have faith in it.