Concentric Hierarchy

The nested set of symmetric polyhedra whose volumes, measured with the tetrahedron as unity, come out as exact integers -- tetrahedron = 1, cube = 3, octahedron = 4, rhombic dodecahedron = 6, vector equilibrium = 20 -- because every form resolves into whole-number counts of A and B Quanta Modules.

Fuller's Definition

Fuller titled the master table "Concentric Domain Growth Rates" (§955.40) and laid out the module counts: the tetrahedron consists of 24 modules, the cube 72, the octahedron 96, the rhombic dodecahedron 144, and the vector equilibrium 480 (§982.02--04). Since each A or B module is exactly 1/24 of a tetrahedron, dividing any module count by 24 yields the polyhedron's volume as a clean integer. The cube's 72 modules give volume 3; the octahedron's 96 give volume 4; the rhombic dodecahedron's 144 give volume 6; the VE's 480 give volume 20 (§982.04). Fuller argued that humanity's adoption of the cube's edge as its dimensional unit forced a "family of irrational constants" for translation, whereas 60-degree coordination yields "omnirational frequency" (§982.12--15).

Significance in Synergetics

The concentric hierarchy is the payoff of tetrahedral mensuration. Where Cartesian geometry produces irrational volumes (the cube-edge tetrahedron has volume 0.9428...), synergetics produces low whole numbers because its unit mesh -- the isotropic vector matrix -- is built from equilateral triangles and the tetrahedron is the natural quantum of volume. The hierarchy is concentric because every form shares a common center in closest packing and nests inside the next: tetrahedron inside cube inside octahedron inside rhombic dodecahedron, all within the VE's 20-ness. This integer accounting extends to the Duo-tet Cube's 24 tetravolumes, which completes the allspace-filling that the VE's 20-ness alone cannot achieve (§1033.701).

See Also

• Vector Equilibrium -- the 20-tetravolume zero-phase reference
• A and B Quanta Modules -- the 1/24-tetra volumetric quanta that produce integer counts
• Isotropic Vector Matrix -- the 60-degree coordinate lattice underlying the hierarchy
• Closest Packing of Spheres -- the physical packing from which these nested forms arise
• Truncated Octahedron -- a related space-filling polyhedron
• Frequency -- modular subdivision that scales the hierarchy
• All-Space Filling -- the 24-tetravolume completion of the VE's 20-ness
• Synergetics -- Fuller's system
• School of Tomorrow / MartianMath (School of Tomorrow / MartianMath) — related topic

Sources

• Synergetics -- Fuller's primary text: §955.40 (Concentric Domain Growth Rates table), §982.00--04 (module counts and whole-number volumes), §982.12--15 (cube edges vs. IVM vectors and irrational constants)
• raw/repos/synergetics.md -- raw source stub