Tetrahedral sphere packing

[wouter meeussen]

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.