## CLOSEST PACKING OF SPHERES

When spheres are packed closely together in concentric layers around a central sphere, certain regularities become apparent. It takes 12 spheres to make up the first layer, 42 for the next, and 92 for the third. Fuller saw that this was the total number of naturally occurring chemical elements, the 92nd in the periodic table of elements being uranium.

He realized that there were structural affinities between his model of closest-packing spheres and the nuclear energy patterns of uranium.

He found that by adding the number of spheres in the three layers, we arrive at the figure 146, which is the number of neutrons in the uranium atom. As he conceives it, the outer layer of an atom system always has an "unemployed associability count": the inner neutrons are surrounded, but the outer shell is associable with an equal number of protons.

Adding this count to the neutron count, we arrive in this case at a total of 238, and this is the actual atomic weight of uranium, i.e. the number of nucleons in the uranium atom.

Fuller established the formula for the number of spheres in any layer: N = 10F^{2} + 2, where 10 is something called the triangular symmetry number, F is the frequency or layer count (whose second-powering Fuller relates to the second-power factors in Newton's gravitational equation and in Einstein's mass-energy equation), and the additive 2 represents the polar spheres, which lie on the axis of spin.

This is discussed concisely in the chapter Energetic Geometry in the Reader (pp.130-132), and in greater depth in Synergetics (410.00).

The matter is pursued in Vector Equilibrium.

American Scientist November-December 1998, Volume 86, No. 6

The Proof Is in the Packing

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THE FULLER MAP

© Paul Taylor 2001