## Figure 15: Equation for Omnidirectional Closest Packing of Spheres

__Equation for Omnidirectional Closest Packing of Spheres__:
Omnidirectional
closest packings of equal radius spheres about a nuclear sphere form
a concentric series of vector equilibria of progressively higher
frequencies. The number of spheres (or, in the case of the counterpart
polyvertexia, the number of edge intervals) on any symmetrically concentric
shell or layer is given by the equation 10F^{2} + 2,
where F = Frequency.
The frequency can be considered as the number of layers (concentric shells
or radius) or the number of edge modules on the vector equilibrium. A
one-frequency sphere packing system has 12 spheres on the outer layer (A)
and a one-frequency vector equilibrium has 12 vertexes. If another layer
of spheres are packed around the one-frequency system, exactly 42
additional spheres are required to make this a two-frequency system
(B), If still another layer of spheres is added to the two-frequency
system, exactly 92 additional spheres are required to make the
three-frequency system (C). A four-frequency system will have 162
spheres on its outer layer. A five-frequency system will have 252
spheres on its outer layer, etc.

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Copyright 1999, Estate of Buckminster Fuller, all rights reserved.