Synergetics (Fuller) - Tetrahedral Accounting

Tetrahedral Accounting

A chief hallmark of this system of mensuration was its unit of volume: a tetrahedron defined by four closest-packed unit-radius spheres. This tetrahedron anchored a set of concentrically arranged polyhedra proportioned in a canonical manner and inter-connected by a twisting-contracting, inside-outing dynamic named the Jitterbug Transformation.

Shape Volume Properties
A,B,T modules 1/24 tetrahedral voxels
MITE 1/8 space-filler, 2As, 1B
Tetrahedron 1 self dual
Coupler 1 space filler
Cuboctahedron 2.5 cb.h = 1/2, cb.v = 1/8 of 20
Duo-Tet Cube 3 24 MITEs
Octahedron 4 dual of cube
Rhombic Triacontahedron 5 radius rt.h < 1, rt.v = 2/3 of 7.5
Rhombic Dodecahedron 6 space-filler, dual to cuboctahedron
Rhombic Triacontahedron 7.5 rt.h = phi/sqrt(2)
Icosahedron ~18.51 edges 1 = tetrahedron's edges
Cuboctahedron 20 edges 1, cb.h = 1
2F Cube 24 2-frequency, 8 x 3 volume
Shape Volume A B T
A module 1/24 1 0 0
B module 1/24 0 1 0
T module 1/24 0 0 1
MITE 1/8 2 1 0
Tetrahedron 1 24 0 0
Coupler 1 16 8 0
Duo-Tet Cube 3 48 24 0
Octahedron 4 48 48 0
Rhombic Triacontahedron 5 0 0 120
Rhombic Dodecahedron 6 96 48 0
Cuboctahedron 20 336 144 0
2F Cube 24 384 192 0

Whole number volumes

A & B modules

Corresponding to Fuller's use of a regular tetrahedron as his unit of volume was his replacing the cube as his model of 3rd powering.(Fig. 990.01) The relative size of a shape was indexed by its "frequency," a term he deliberately chose for its resonance with scientific meanings. "Size and time are synonymous. Frequency and size are the same phenomenon." (528.00) Shapes not having any size, because purely conceptual in the Platonic sense, were "prefrequency" or "subfrequency" in contrast.

Prime means sizeless, timeless, subfrequency. Prime is prehierarchical. Prime is prefrequency. Prime is generalized, a metaphysical conceptualization experience, not a special case.... (1071.10)

Generalized principles (scientific laws), although communicated energetically, did not inhere in the "special case" episodes, were considered "metaphysical" in that sense.

An energy event is always special case. Whenever we have experienced energy, we have special case. The physicist's first definition of physical is that it is an experience that is extracorporeally, remotely, instrumentally apprehensible. Metaphysical includes all the experiences that are excluded by the definition of physical. Metaphysical is always generalized principle.(1075.11)

Tetrahedral mensuration also involved substituting what Fuller called the "isotropic vector matrix" (IVM) for the standard XYZ coordinate system, as his principal conceptual backdrop for special case physicality:

The synergetics coordinate system -- in contradistinction to the XYZ coordinate system -- is linearly referenced to the unit-vector-length edges of the regular tetrahedron, each of whose six unit vector edges occur in the isotropic vector matrix as the diagonals of the cube's six faces. (986.203)

The IVM scaffolding or skeletal framework was defined by cubic closest packed spheres (CCP), alternatively known as the FCC or face-centered cubic lattice, or as the octet truss in architecture (on which Fuller held a patent). The space-filling complementary tetrahedra and octahedra characterizing this matrix had prefrequency volumes 1 and 4 respectively (see above).

A third consequence of switching to tetrahedral mensuration was Fuller's review of the standard "dimension" concept. Whereas "height, width and depth" have been promulgated as three distinct dimensions within the Euclidean context, each with its own independence, Fuller considered the tetrahedron a minimal starting point for spatial cognition. His use of "4D" was in many passages close to synonymous with the ordinary meaning of "3D," with the dimensions of physicality (time, mass) considered additional dimensions.

Geometers and "schooled" people speak of length, breadth, and height as constituting a hierarchy of three independent dimensional states -- "one-dimensional," "two-dimensional," and "three-dimensional" -- which can be conjoined like building blocks. But length, breadth, and height simply do not exist independently of one another nor independently of all the inherent characteristics of all systems and of all systems' inherent complex of interrelationships with Scenario Universe.... All conceptual consideration is inherently four-dimensional. Thus the primitive is a priori four-dimensional, always based on the four planes of reference of the tetrahedron. There can never be less than four primitive dimensions. Any one of the stars or point-to-able "points" is a system-ultratunable, tunable, or infratunable but inherently four-dimensional. (527.702, 527.712)

Synergetics did not aim to replace or invalidate pre-existing geometry or mathematics, was designed to carve out a namespace and serve as a glue language providing a new source of insights.

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