## The Problem of the Thirteen Spheres

When one gathers 13 equal-radius spheres (marbles, ping-pong balls, ball bearings, BBs, whatever) into one's hand and forces them into their closest possible "packing," one ends up with 12 balls around and in contact with 1 ball in the center.
This is "sphere-approximate" (it is basically a "nucleated" icosahedron of balls).

One further finds that, unless one has by chance fit them into a "cuboctahedral" arrangement, the 12 balls on the outside of this are somewhat movable. That is, while holding them in the hands, one can move each ball slightly, as if one had a sort of a semi-spherical 3D puzzle.

One may also twist this collection in various ways, as the balls move against each other (it helps if they are oiled).
Possible forms include the regular icosahedral form (no outside ball touches another outside ball), the "cuboctahedron" arrangement (Buckminster Fuller used this arrangement extensively), and an arrangement in which the greatest number of outside balls touch (this looks "lopsided," is what I call the "lopsided icosahedro(not)," or "Our Lady in Jade," and is the center and core of all packing-reasoning in Tetrahedraverse).

There are, in addition to these clearly-definable (hence labellable) ones, probably an infinite number of intermediate arrangements for those 12 balls.

Why should the 13 not pack tightly in all cases except one out of a practical infinity of cases?
Why is there this "sloppiness" in the outside 12 balls?

This is "the problem of the 13 spheres."

Tetrahedraverse is in essence an extended (but not quite infinitely) packing of dimensionless pionts (points) in a gigantic sphere the size of the universe, which, by virtue of this 12-around-1 packing being the tightest possible for any 13 equal spherical regions that behave like frictionless balls, is patterned like a Network of Vertex Order Twelve.

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