The Synergetic Temperament System

Kenneth Hemmerick

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The Closest-packing of Spheres (Nodal Points)

In a 2-dimensional wave formation, nodal points are packed extending in two directions - one positive, the other negative. In a 3-dimensional wave formation, nodal points are closest-packed, extending in 12 directions. In a 2-dimensional wave formation, the number of nodal points outside the innermost node defines the frequency of the wave. In a 3-dimensional wave formation, the number of points found in the outer-shell defines the frequency.

The geometric configuration formed by the closest-packed layering of spheres around a central sphere of the same magnitude is called the vector equilibrium or cubo-octahedron. The vector equilibrium is a polyhedron bounded by eight triangular faces and six quadrangular faces.


The name (vector equilibrium) is derived from the fact that the radial vectors of this figure have the same value as the circumferential vectors. In terms of dynamics, the outward thrust is exactly balanced by the restraining chordal force; thus the figure is an equilibrium of vectors.²


The number of spheres needed to enclose a central sphere of the same size is 12. If the same omni-directional closest-packed layering in continued, 42 spheres will enclose the 12, 92 will layer the 42, 162 will layer the 92 and so forth. The formula used to describe this behavior is found in the Energetic Synergetic Geometry of R. Buckminster Fuller.³ It reads as follows:

P = 2 + (2*5) X (F)²

The number of points (P) in the outer layer (shell) of any symmetrical system (vector equilibrium), is 2 plus 2 times 5, multiplied by the system's edge frequency to the second power. 3

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© 2006 Kenneth Hemmerick