The Single Local Rule of Ffellonics

One rule, twelve levels

Ffellonics makes a deliberately spare claim: a single local rule, applied over and over to identical units, is enough to generate the whole Ffellonic Hierarchy. That hierarchy is a twelve-level developmental progression, running from the first contact between two spheres to a maximally coordinated lattice. Nothing is added along the way. No new instruction switches on at Level 6, and no plan of Level 12 is stored anywhere in Level 1.

This essay sets out that rule, explains why its locality matters, and shows how far it carries the hierarchy. It also marks where the rule is under strain, credits those who reached parts of it first, and says what evidence would count against it. The aim is not to make the rule look larger than it is. The aim is to show that, stated carefully, it is a strong and testable idea.

The rule stated

The rule is: symmetric nearest-neighbour attachment under free-energy minimisation, up to each sphere's binding capacity. Each new unit joins the existing structure where it makes the most contacts with neighbours it can touch, without exceeding the number of bonds a sphere can hold, in the arrangement that best preserves symmetry, so that the system's Gibbs free energy falls. Each part of that sentence does specific work.

Identical units. The rule applies to units that are indistinguishable: same size, same surface, same interactions. In the physical baseline these are spheres with an isotropic attraction. The range of that attraction is a parameter of the rule's conditions, alongside temperature and binding capacity, not fixed in advance. Identity matters because it removes any information that could steer the outcome. Whatever order appears must come from geometry and thermodynamics, not from labels on the parts.

Nearest-neighbour. A unit responds only to what it touches. There is no action at a distance and no sensing of the whole. This is what makes the rule local.

Contact-maximising. Every contact lowers the energy of the system. Other things being equal, a site offering three contacts is preferred to one offering two. This is the engine of the progression, since coordination can only rise as contacts accumulate.

Symmetric. Among placements with equal contacts, the rule favours the most symmetric arrangement. This is the most distinctive clause of the rule, and also the one most open to challenge, as the section on tensions explains.

Under free-energy minimisation. The governing quantity is Gibbs free energy, G = H − TS, at constant temperature and pressure. Contacts lower the enthalpy H. But entropy S also counts: a rigid, highly symmetric cluster can lose out to a floppier or less symmetric one at finite temperature. Thermodynamics sets the direction of assembly and the total dissipation; kinetics, meaning supply, diffusion and energy barriers, sets its pace.

Up to binding capacity. A sphere can hold only so many bonds, and that capacity rises as the hierarchy progresses, so that more spheres can be attracted to each sphere until Level 12. At each level the rule builds the arrangement that fills that level's capacity: four bonds per sphere selects the octahedron, five the hollow icosahedron. At twelve, the most neighbours an equal sphere can have, the cap no longer limits anything, and the rule becomes ordinary isotropic attraction, which ends in close packing. The companion essay, Ffellonics: Range, Temperature and Binding Capacity, sets out the calculations behind this clause.

Put together, these clauses describe what the Ffellonic Hierarchy is: a developmental progression that moves in one direction only. That direction is towards higher coordination, under the conditions the rule assumes.

What the rule leaves out

The force of the rule lies as much in its absences as in its content. Three things are missing, and each absence is a claim.

No blueprint. The final lattice is not specified anywhere in advance. Level 12 is not a target the units aim for; it is where the rule's repeated application ends up when nothing interferes. In the language of biology this is teleonomy, not teleology: the process is directed, but not by a goal held in mind.

No coordinator. No unit, and no outside agent, oversees the whole. Order at the scale of the cluster is produced by decisions made at the scale of a single contact. This places Ffellonics in the family of self-organising systems, alongside crystal growth and the self-assembly of colloids.

No change of rule. The same rule operates at every level. The hierarchy does not need a second principle to get from the tetrahedron to close packing. What changes from one level to the next is the structure the rule acts on and the binding capacity of the spheres, not the rule itself.

This economy is the main attraction of the framework. If one short rule can account for a long ordered sequence, the sequence is explained rather than merely described.

How the hierarchy unfolds

Applied repeatedly, the rule produces a sequence of structures of rising coordination. Before Level 1 there are only units without relation: pure potential. The milestones then run as follows.

Level 1: the dyad. Two spheres touch. Each has one neighbour; this is the first relation.
Level 2: the triangle. A third sphere touches both. Each sphere now has two neighbours, in the first closed figure.
Level 3: the tetrahedron. A fourth sphere sits in the hollow above the triangle. Each sphere touches the other three. This is the first rigid three-dimensional form and the basic cell of dense packing.
Level 4: the octahedron. Six spheres, each touching four neighbours, with twelve contacts in all. Among six-sphere arrangements, the rule's symmetry clause selects this one.
Level 5: the hollow icosahedron. Twelve spheres on a shell, each touching five neighbours, thirty contacts in all, with nothing at the centre. It is an important milestone in nature, most famously in viral capsids.
Levels 6 to 11: further coordination shells. Each level raises by one the number of neighbours a sphere touches.
Level 12: close packing. Each sphere touches twelve others. Twelve is the kissing number in three dimensions, the greatest number of equal spheres that can touch a central one.

Level 12 is best described as a class of maximally coordinated lattices, not a single ground state. Face-centred cubic (FCC) and hexagonal close-packed (HCP) stacking both give every sphere twelve neighbours and the same density, about 74%. Kepler conjectured that no packing of equal spheres is denser; Thomas Hales proved it. Which of the two stackings a real system adopts depends on effects beyond the nearest neighbours, so the rule by itself leaves them tied.

The numbering follows a single principle: a structure's level is the number of neighbours each sphere touches. That is why the octahedron is Level 4 and the icosahedron Level 5, and why the progression ends at twelve. Each step has more contacts per sphere than the last, and none is reached by abandoning the rule.

Range acts on every level, not on one. Levels 1 to 3 have no competitor: for two, three and four spheres the contact-maximising structure is unique, so it forms at any range. Six spheres is the first point where two structures tie on contacts, so it is the first point where range and temperature visibly decide the outcome. Longer range and lower temperature favour the octahedron at Level 4. Longer range also favours icosahedral order, while shorter range destabilises strained structures and pushes finite clusters towards close packing (Doye, Wales and Berry 1995). In the bulk, every range ends in a twelve-coordinated lattice at Level 12.

Where the rule is under strain

A single rule invites a single question: does it really produce every level, and only those levels? Honest answers show several points of strain. Two of them, at six and twelve spheres, are resolved once range, temperature and binding capacity are stated as conditions; the companion essay gives the calculations. The others limit what can be claimed.

Five spheres are not a named level. At N = 5 the rule produces a structure (a triangular bipyramid of two tetrahedra), but the hierarchy does not single it out. The sequence of named milestones is therefore a selection from what the rule generates, not a complete list of it.

At six spheres, the outcome depends on the conditions. The octahedron and the polytetrahedron (three tetrahedra sharing faces) both have twelve contacts. With very short-range attraction they tie in energy and entropy decides: a highly symmetric cluster carries less rotational entropy, so the polytetrahedron is the more common outcome. Meng, Arkus, Brenner and Manoharan measured exactly this in colloids (Meng et al. 2010).

Two conditions restore the octahedron. With longer-range attraction its next-nearest neighbours sit closer, so it becomes the lower-energy structure and wins at low temperature; at very long range the polytetrahedron is not even stable (Doye, Wales and Berry 1995 on range generally; the six-sphere numbers are the companion essay's own calculation). With a binding capacity of four, the octahedron makes all twelve possible bonds while the polytetrahedron, which has two five-neighbour spheres, is trimmed to eleven. Under the conditions of Level 4, the octahedron wins on energy.

The icosahedron needs its binding capacity. For isotropic spheres with unlimited binding, the hollow icosahedron never wins: a close-packed fragment of twelve spheres makes 33 contacts to its 30 (Arkus, Manoharan and Brenner 2009), and its high symmetry also costs it entropy. With a binding capacity of five, the result reverses. Thirty bonds becomes the ceiling, the hollow icosahedron alone reaches it, and close-packed fragments manage at most 28. Level 5 is therefore the most stable configuration under its own condition, not under isotropic attraction. Frank (1952) showed why icosahedral order is locally favoured in simple liquids, which is also why it can delay crystallisation.

Some regular forms never appear. The cube and the dodecahedron are not generated. Their arrangements have too few contacts and are floppy, so a contact-maximising rule passes them by. This is a consequence of the rule, not a flaw, but it means the hierarchy is not a tour of all the Platonic solids.

Levels are not guaranteed minima. The named levels are milestones on a route, not each a proven free-energy minimum for its number of spheres. Any fixed-N comparison must count floppy-mode and symmetry-number entropy, not energy alone.

Equilibrium is not pathway. The rule is stated as cumulative attachment, each contact locked in as it forms. The evidence for it, from the cluster literature and the companion calculations, concerns equilibrium: which arrangement is most stable when bonds can break and re-form. The two agree only when assembly is slow, reversible or annealed. Where barriers are high, clusters can be trapped in whatever formed first, and less stable forms often appear before the most stable one. Thermodynamics sets the destination; kinetics decides whether and how it is reached, and the kinetics of the hierarchy is not yet worked out.

Taken together, these points show that the symmetry clause does its work through the conditions, not alone. Where range, temperature and binding capacity are stated, the contested cases fall into place; where they are not, the contact-maximising clause is on firmer ground than the symmetry clause.

Prior art and what is new

The rule stands on a long line of work, and its case is stronger for saying so.

Kepler (1611) conjectured that close packing is the densest arrangement of equal spheres. Hales's proof settled it centuries later, giving Level 12 its mathematical floor.
Frank (1952) explained why local icosahedral order is favoured in simple liquids, which is the source of the icosahedral tension above.
Buckminster Fuller built his synergetics on the isotropic vector matrix: the network of equal spheres in closest packing, seen as a frame of equal vectors. Ffellonics shares his starting point and his interest in how form grows from contact.
The sticky-sphere literature is the closest scientific neighbour. Arkus, Manoharan and Brenner (2009) enumerated the ground states of small clusters of spheres with short-range attraction. Meng et al. (2010) measured their free energies in experiments. Holmes-Cerfon, Gortler and Brenner (2013) showed that, as the range of attraction shrinks, the free-energy landscape becomes purely geometric plus one control parameter. Kallus and Holmes-Cerfon (2016) found that clusters with extra contacts dominate the landscape, and that because these resemble fragments of close-packed lattices, order can emerge in clusters as small as ten spheres.

That last result is the strongest outside support for the contact-maximising clause: in a real statistical-mechanical treatment, contact-rich, close-packing-like clusters do win, and early.

What Ffellonics adds is not new physics at the level of individual clusters. It is an organising claim: that one rule, read cumulatively, threads these results into a single ordered progression with a fixed start and a class of end points. It also adds a use. Treated as a reference model, the progression gives a baseline against which departures can be read.

What would count against the rule

A rule worth holding is a rule that could fail. These findings would weaken or overturn it:

Systems of identical units with isotropic attraction, free to rearrange, that reliably settle into low-contact or non-close-packed structures at equilibrium, not merely because they are kinetically trapped.
Evidence that the predicted structure is systematically not formed at equilibrium where the stated range, temperature and binding capacity say it should be: for example, five-capacity particles that settle into something other than hollow icosahedra, or long-range six-sphere clusters that settle into polytetrahedra at low temperature.
A missing step: a named level that is not the equilibrium structure at any setting of range, temperature and binding capacity.

Other departures are not counter-evidence. They are what the reference model is for. Strongly directional or heterogeneous units, assembly driven by entropy alone, and systems that arrest into glasses fall outside the rule's stated conditions; when they produce different structures, the difference tells us which condition was broken. Particles with a limited number of binding patches are a different case: they are the rule's binding-capacity condition made real, and so a direct test of it (Wang et al. 2012).

Conclusion: a rule as a baseline

The single local rule is best understood the way physicists understand the ideal gas law or biologists the Hardy–Weinberg principle. It is not a claim that every system obeys it. It is a claim about what identical units, attracting with a given range at a given temperature and holding a given number of bonds, will do when left free to rearrange, and a standard against which real systems can be measured.

Read that way, the rule is both modest and powerful. It is modest because it holds only under stated conditions, ties FCC and HCP at its summit, and speaks to the destination of assembly, not yet its pathway. It is powerful because, once range, temperature and binding capacity are stated, the cases that looked like counter-examples become predictions: the octahedron where binding is four-fold or long-ranged, the hollow icosahedron where it is five-fold, and close packing where it is unlimited. Independent work on sticky spheres, cluster structure and close-packed metals broadly confirms the direction it predicts.

That is the heart of Ffellonics. Order does not have to be imposed from above. Under the right conditions, it accumulates one contact at a time, along a developmental progression that moves in one direction only.

References

Checked against the source pages for this essay:

Arkus, N., Manoharan, V. N. and Brenner, M. P. (2009). Minimal energy clusters of hard spheres with short range attractions. Physical Review Letters 103, 118303.
Meng, G., Arkus, N., Brenner, M. P. and Manoharan, V. N. (2010). The free-energy landscape of clusters of attractive hard spheres. Science 327, 560–563.
Holmes-Cerfon, M., Gortler, S. J. and Brenner, M. P. (2013). A geometrical approach to computing free-energy landscapes from short-ranged potentials. PNAS 110(1), E5–E14.
Kallus, Y. and Holmes-Cerfon, M. (2016). Free energy of singular sticky-sphere clusters. arXiv:1605.08678.
Doye, J. P. K., Wales, D. J. and Berry, R. S. (1995). The effect of the range of the potential on the structures of clusters. Journal of Chemical Physics 103, 4234–4249.
Wang, Y. et al. (2012). Colloids with valence and specific directional bonding. Nature 491, 51–55 (volume and pages from memory).
Fell, D. (2026). Ffellonics: Range, Temperature and Binding Capacity. Companion essay, with the calculations referred to here.

Standard sources cited from memory, not re-checked here:

Kepler, J. (1611). Strena seu de nive sexangula (On the six-cornered snowflake).
Hales, T. C. (2005). A proof of the Kepler conjecture. Annals of Mathematics 162, 1065–1185.
Frank, F. C. (1952). Supercooling of liquids. Proceedings of the Royal Society A 215, 43–46.
Fuller, R. B. (1975). Synergetics: Explorations in the Geometry of Thinking. Macmillan.