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

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

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

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.)

(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)

However,

Moreover, the Hales and Ferguson proof of Kepler's Conjecture should not be mistakenly, generically, applied to equi-radius ball-packings of

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