A Possible Counter-example to Hales and Ferguson's
Proof of the Kepler Conjecture

The Cubic Close Packing ("CCP") of equal-radius spheres is considered the densest possible packing, having a density of nearly 74 percent. However, a distinction needs to be made between the success of Hales and Ferguson's proof of that position when considering an arbitrary number of spheres, and packings with a lesser number of spheres.

For example, my packing of 13 spheres ONLY, called by me either an "icosahedro(not)" or "Our Lady in Jade" ("...Bald; Forever Stainless" ...don't ask), or by others "teticosa9" (cf. John Braley) and "weary icosa," beats the CCP packing slightly.

If one removes a 13-closest spheres chunk from an extended CCP, one gets a "cuboctahedron" of spheres. Connecting the centers of these spheres produces what Buckminster Fuller called a "Vector Equilibrium" ( ' VE ' ). Steve Waterman's polyhedra are obtained in a similar manner, by extracting spherical regions from within a CCP, and at the center of every Waterman Polyhedron is a Vector Equilibrium of spheres (but, Steve prefers his own name for this, the "Root 1 Waterman").

I discovered (or, RE-discovered; I have few illusions about my own genius) the semi-loose, 'lopsided' arrangement of 13 equal radius spheres by "packing" spheres the same way Fuller instructs to pack, but I was unaware of his instructions in those years (late 1980s), so I did not know I was "supposed to" end up with a VE. (Lucky me; no preprejudices.)

Now, over the years, and eventually, upon discussions with Fuller-Followers, I found a peculiar and misleading prejudice existed in the Fuller camp, that no other packing could be denser than the 13-sphere VE-pack because "everyone knows" that the CCP, from which the VE may be extracted over and over again, is the tightest/densest possible.

This prejudice even led Mr. Waterman, who is not a Fuller-Follower although he respects Fuller's work as much as do I, to believe that because his packings were extracted from the CCP, no other arrangement could be tighter even though he was extracting spherical regions from it. (This is what led to an online "challenge" between one of Steve's packings and one of mine, a packing challenge that I won with my piont-packing program.)

Density of the 13-ball packing; attempts-to-prove

First, know this: when enclosed in a tangent container-sphere three ball-radii in radius, all nucleated 13-ball packings have identical density.

This should be obvious, since the density (space occupied by balls divided by total space in the container-sphere) is calculated using the same radius container-sphere and the same 13 balls regardless of how they are placed in the container-sphere. The difficulty in challenging Hales/Ferguson over a 13-only ball pack arises when trying to determine density of the pack in isolation —not within any container, spherical or otherwise.

Here are examples of various placements of the 13 balls within the container sphere. There are in theory an infinite number of possible such arrangements:

(X-eyed stereo): Top to bottom 1) Regular icosahedral (note no outside ball touches another); 2) My icosahedro(not); 3) Cuboctahedral (Fuller's 'VE'). (Images not to scale.)
13 ball packs image
(Image copyright © 2004 by John Brawley)

(Every outside ball —each of the 12 around the central one— in each of these is in contact with the central ball.)

We (self, and correspondents from "Synergeo" mailing list) tried these methods:

* The Hulled-facets (polyhedral area) argument (images)
* The 20-tetrahedra-volume argument (Struck-to-PovRay images)
* The Perimeter argument, including the Argument from Gap (no images)
* The Center-of-Density (diagonals-missing-center) argument (Struck/PovRay images) and, related,
* The Opposite-pairs (diagonals too short) argument (Struck images)
* The "hang-it-from-a-tripod" argument (SpringDance image)
* The PackingMink argument (PackingMink link, and image)
* The QHull Voronoi Cell argument (QHull link, image)

UpShot !

Nobody knows for absolutely sure. Use the above info to draw your own conclusion.

However, _I_ claim my icosahedro(not) ("Our Lady in Jade; Bald, Forever Stainless") is the tightest/densest possible sphere-packing for 13 spheres ONLY, and that there is no way that Fuller's 'VE', nor any other arrangement, beats my configuration for density.

Moreover, the Hales and Ferguson proof of Kepler's Conjecture should not be mistakenly, generically, applied to equi-radius ball-packings of any number of balls. It most likely does not apply to very low-ball-numbered packings such as this one.

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