Frequency (Synergetics)

The number of modular subdivisions of a system's edges -- Fuller's way of expressing size not as absolute length but as how finely a structure is subdivided, so that a 2-frequency geodesic has twice as many edge modules as a 1-frequency.

Fuller's Definition

Fuller defined frequency as "the number of spaces between the balls, not the number of balls" in closest packing (§515.31), and more formally as the count of modular subdivisions of a system's radial or circumferential vectors (§515.011). "The areal and volumetric quanta of separately islanded 'points' are always accountable numerically as the second and third powers of the frequency of modular subdivision of the system's radial or circumferential vectors" (§515.011). Angles are "subsize" and generalized; "the length of the edges is frequency, while the angle is subfrequency" (§515.14). Man needs "only generalized angles and special-case frequencies to describe any and all omnidirectional patterning experience" (§515.12). In closest packing, each concentric shell contains exactly 10F^2 + 2 spheres, where F is the edge frequency (§222.01--03). The frequency is also the number of concentric layers: "the number of layers and the number of edge modules is the same" (§222.03).

Significance in Synergetics

Frequency is Fuller's bridge between the timeless, sizeless generalized principle and the special-case physical artifact. Triangles and tetrahedra are "thinkable independently of size" because their angles are subcyclic constants (§515.101, §515.11); frequency is what makes them physical -- each cyclic sizing increment "inherently constitutes one unit of experienced physical energy" (§515.11). A geodesic dome's complexity is described entirely by its frequency: a 2-frequency icosahedron has its edges halved, a 3-frequency has them thirded, each higher frequency yielding more triangles and closer approach to a sphere. The formula 10F^2 + 2 governs not only shell counts in closest packing but also the vertex count of any frequency-specified vector equilibrium, connecting frequency directly to the concentric hierarchy.

See Also

• Closest Packing of Spheres -- shell counts governed by 10F^2 + 2
• Vector Equilibrium -- the form whose frequency-layers the formula describes
• Isotropic Vector Matrix -- the matrix whose edge modules define frequency
• Concentric Hierarchy -- the volume ratios frequency scales
• All-Space Filling -- higher-frequency tessellation
• Synergetics -- Fuller's system

Sources

• Synergetics -- Fuller's primary text: §222.01--03 (10F^2 + 2 sphere-shell formula, frequency = edge modules = layers), §515.00--33 (frequency definition, angles as subfrequency, energy), §610.20--24 (three prime structural systems at increasing frequency)
• raw/repos/synergetics.md -- raw source stub