Tetrahedral sphere packing
Jonathan Post
jvospost3 at gmail.com
Wed Jan 2 00:16:42 CET 2008
Technically, are these packings of TETRAHEDRAL SPHERES -- the central
projections of the tetrahedron onto the surface of the unit sphere?
The triangular sides of the tetrahedron become spherical triangles on
the surface of the sphere.
This would then be quite different from enumerations of a subset of A119602?
On 1/1/08, wouter meeussen <wouter.meeussen at pandora.be> wrote:
> best of wishes to you all,
>
> something cute for you to ponder:
> imagine a deterministic sphere packing, starting from a single tetrahedron
> of spheres.
> Now, layer by layer, add spheres to all triangular faces where this can be
> done avoiding overlap.
>
> Don't be fooled by the presumption that this technique will lead to a
> *dense* packing like the BCC or FCC packing with their nice 74% density. No
> way! It doesn't even approach the 64% density of the random sphere packing.
> Reason: the technique above restricts placement to sites where a new sphere
> touches three *mutually touching* spheres. More like 47%. Boo!
>
> The only downside to the deterministic nature of the technique is:
> the sequence in which candidate faces are adorned with new spheres is as yet
> undefined. This looks a bit like visiting nodes on a graph (depth first or
> width first). Analogously, we can either fully adorn each candidate
> tetrahedron (triple of empty faces) with new spheres, and then move on to
> the next, or, we can adorn each tetrahedron with one sphere each, and then
> return to the first and try to put a second sphere on each, then finally a
> third on each. (Did you catch the christmas tree analogy here?).
>
> The counts of tetrahedrons (= count of spheres - 3) generated are slightly
> different:
> depth first:
> 1,5,11,19,28,37,48,64,82,103,127,160,198,238,283,340,411,482,564,645,735,830
> ,942,1071, ..
> width first:
> 1,5,11,21,29,40,54,70,85,106,133,167,205,247,290,349,404,469,548,636,725,828
> ,944,1068, ..
> (nothing via superseeker).
>
> I withhold submission until a later date, so that knowledgeable folk can
> grunt at this.
> I've put the Mathematica 4.0 version of this program on
> http://users.pandora.be/Wouter.Meeussen/SphereTetrahedralPacking1.nb
> it's 719 kilobytes.
>
> W.
>
>
Wouter,
I worked on that problem in 1995. Take a look at
#193 on my publication list:
<LI><STRONG><A NAME="P10193">193. <A></STRONG><STRONG>Minimal-Energy Clusters of Hard Spheres</STRONG>
[<a href="duff.txt">Abstract</a>, <a href="duff.ps">postscript</a>, <a href="duff.pdf">pdf</a>],
[note:
<a href="dufffig11.ps"> Fig. 11 </a>,
<a href="dufffig12.ps"> Fig. 12 </a>
and "photos" of the putatively optimal clusters of
<a href="clust4_10.gif">4 to 10</a> and
<a href="clust13_20.gif">13 to 20</a> spheres
are in separate files]
N. J. A. Sloane, R. H. Hardin, T. S. Duff and J. H. Conway,
<EM>Discrete Computational Geom.</EM>,
14 (1995),
pp. 237-259.</LI>
Well, it's not the same as your problem, but yours
is really not well defined. Our way makes it into
a much cleaner problem.
Neil
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