The hierarchy of ffellonic geometry ranks all regular geometric forms

Ffellonic geometry is a geometric framework (introduced around 2000 by David Fell) that reinterprets regular 3D forms through the lens of identical spheres packing symmetrically in 3D Euclidean space. It emphasizes natural attachment rules: spheres connect to nearest neighbors while maximizing contacts (coordination number k), preserving symmetry, and minimizing energy/strain.

Unlike the traditional static view of the five Platonic solids (tetrahedron, cube, octahedron, dodecahedron, icosahedron) as an isolated complete set, ffellonic geometry embeds them within a progressive, hierarchical sequence. This hierarchy ranks structures by increasing complexity, coordination, and order — from minimal pairwise contacts up to the maximum possible in 3D space (k=12 in dense packings like FCC/HCP).

The core idea is a single generative rule leading to a clean 12-level hierarchy, where each level builds on the previous through symmetric sphere additions. Spheres act as vertices, contacts as edges, and the progression reflects nature's preference for low-energy, dissipative structures (e.g., analogous to crystal growth or self-assembly).

Key Aspects of the Hierarchy

• Starts at Level 1: simple dyad (two spheres touching, k=1 coordination per sphere).

• Builds through clustered symmetric arrangements with increasing nearest-neighbor contacts.

• Incorporates the Platonic solids (and their duals) as key intermediate stages, but redefines them dynamically.

• Extends beyond finite polyhedra into lattice-like or spaceframe structures.

• Culminates at Level 12: maximum symmetric coordination (k=12) in regular dense packings (face-centered cubic or hexagonal close-packed), where every sphere touches exactly 12 others.

This creates a ranked ordering of "regular" 3D geometric forms based on:

• • Increasing coordination number (k = 1 to 12)

• • Growing symmetry and stability

• • Natural emergence from sphere-based packing rather than purely face-based polyhedral definitions

Some sources describe partial or extended lists of "ffellonic forms" (e.g., straight line → triangular polygon → tetrahedral → octahedral → icosahedral → hexagonal tessellation → linear truss → octahedral spaceframe → and further levels toward full lattices), but the overarching model consistently uses the 12-level progression as the ranking system.

In short, ffellonic geometry provides a ranked hierarchy of symmetric 3D regular forms by treating them as stages in symmetric sphere packing, unifying Platonic solids with broader packing phenomena into one coherent, energy-driven ladder up to the densest possible regular arrangements in 3D. This approach has been explored for analogies in quantum computing, quantum gravity, materials science, and natural pattern formation.