This project investigates the maximum achievable energy density in a directional bond network built on an icosahedral ray grid.
The study is purely geometric and topological. Although the motivation comes from atomic nucleus packing, the simulation deliberately ignores nuclear physics and focuses only on:
The goal is to determine how efficiently bonds can populate a predefined directional grid. The rules are created based on the Phase-locked resonance Ontology, the Rotonal Quantum Model and inspired by the Olavian Atom Model.
This sections show a few glimpses of some of the simulation runs.
What this simulation already reveals:
At its core, the model is a directional bond network with icosahedral geometry. The investigations into the Rotonal Quantum Model showed up that geometrical structure of established resonance channels and the identified 3-channels per Proton/Neutron and 12 channels per Alpha-Particle. Create a icosahedral structure where the rotating base elements create Quon-Disks. Rotation-planes which correspond to the faces of a dodecahedron. From this on we’d like to give a prediction of how energetically optimal nuclear cores are built.
Nodes are points in space that carry a local icosahedral orientation frame. In RQM this would be an resonant electron-node or in the simplified model-view a rotonal quon. We will see that we have “steady” nodes which allow for externally extending resonance-channels (mostly electron-span channels) and we will have resonant/rotonal nodes which will undergo rotonal resonances. The later can not expose electron-span channels anymore, due to their higher level rotation in which they are involved. This phase-locked self-closing resonance-loops are closed and can not realize any stable external channels on that span. The still can on their new rotonal span (quon). Which they effectively do. This will create resonance-chains through the nucleus core.
Each node defines:
A bond forms when two nodes meet along opposite rays. Each bond contributes: $Ep / 3$ energy. This will later be mapped to the energy level of a Neutron and/or Proton. So Ep/3 will effectively partly related to the standard physics quark construct. In contrast RQM further differenciates the substructures of up and down quarks. Relating it more to core geometry, core dynamic and energy density rather than secondary effects like charge/coulomb attraction/repulsion.
Nodes themselves contribute negligible energy, so the system is bond dominated.
The network grows by:
Maximize energy density, which is equivalent to maximizing: bond density per volume because each bond carries the same energy.
Because icosahedral symmetry cannot tile Euclidean space perfectly, the network inevitably develops:
The simulation aims to quantify these effects.
The system we aim for should also have these constraints:
Which geometrical figures allwo this:
Even if we do not force any extra penalties, this will produce:
So by using icosahedral structure might automatically optimize for symmetry if we optimize for density
The optimization target lies on energy density. This though is not only a global aspect. Energy density is also a local aspect that accumulates with a uniform gradient outward. How can this be implemented?
Existing concepts are “energy density field smoothing” or “minimize gradients of energy density”, “surface tention models”.
For the moment we will continue though with the symmetry and seceond moment isotropy approach as this better alings with external symmetry.
The directions correspond to the vertices of the icosahedron and can be written using the golden ratio:
(0, ±1, ±φ)
(±1, ±φ, 0)
(±φ, 0, ±1)
This produces the 12-ray star around each node.
Five-fold symmetry conflicts with translational symmetry.
Result:
This is the origin of geometric frustration.
In the Olavina Atom Model this gives a basis for the prediction of stability and maximal size of atom cores. Or in invers way, give information on how big the angular range for “fully entanglement channels” can be.
Each node has:
This is analogous to chemical valence models.
But instead of orbital geometry, the directions come from icosahedral symmetry. In the case at hand this model is now considered as a possible candidate for nuclear bindings via resonance channels.
Two layers appear:
Which bonds exist?
How much distortion is needed to embed the network in real space?
Separating these simplifies simulation design.
Most physical simulations minimize total energy.
This model maximizes:
bond density per volume
That emphasizes packing efficiency of directional bonds.
Several components already exist in known physics.
Metal clusters often adopt icosahedral structures.
Examples include Mackay clusters with magic numbers:
13, 55, 147, 309
These correspond to successive shells.
A major topic in condensed matter physics.
Occurs when: local order ≠ global tiling
Examples:
Particles with directional bonding sites used in simulations of:
The rays we are going to simulate here correspond closely to patch directions.
Used in systems such as:
Focus is on:
rather than pair potentials.
Most real systems have 3–6 bonding directions.
Examples:
system valence
carbon 4 silicon 4 water 4 simple crystals 6
A 12‑direction valence system is unusual. Even though we are aware, that in an icosahedral lattice not all 12 directions will be establish-able for most clusters.
Most models optimize:
Here the focus is: bonds per volume which is closer to a packing efficiency problem.
Nodes sit on a directional scaffold rather than emerging from pair potentials. This makes the system partly graph‑theoretic.
Why is this chosen for RQM simulations:
Key concept: local order incompatible with global space
Important authors include:
Particles with directional bonding sites used to model:
Important researchers:
Classic work by:
Focus areas:
Icosahedral symmetry appears naturally in quasicrystals.
Key concept: long‑range order without periodicity
Important figures:
Valence networks studied in materials like:
Important theory: Phillips–Thorpe rigidity theory.
The 12‑ray star is closely related to spherical coding theory, which studies optimal distributions of directions on a sphere. This connection may help predict the coordination numbers and defect distributions observed in simulations.
The key question becomes:
How much bond density can an icosahedral directional network sustain before frustration forces defects?
Possible outcomes include:
The entire discussion reduces to a geometric question:
What is the densest directional bond network compatible with icosahedral symmetry in Euclidean space?
This sits at the intersection of:
Nodes are points arranged on a predefined icosahedral grid.
Each node carries an icosahedral frame defined by the 12 vertex directions of an icosahedron.
These directions are called rays.
node
→ 12 rays
→ each ray may form a bond
Each ray represents a potential connection direction.
Rays correspond to the 12 vertex directions of an icosahedron.
Properties:
Two opposite rays form a ray axis.
6 axes
12 rays
A bond is formed when two nodes connect along opposite rays.
node A —- bond —- node B
Properties:
Not all rays necessarily lead to valid neighbors in the grid.
Therefore nodes may have:
Energy is stored almost entirely in bonds.
Each bond contributes
Ep / 3
to the total energy.
Nodes may carry a very small intrinsic energy
p < 0.1% Ep
This contribution can be neglected for the initial analysis.
Because icosahedral symmetry cannot tile Euclidean space perfectly, the grid will contain frustration.
Frustration appears as:
The simulation will track these defects explicitly.
The network will be constructed bond by bond.
Procedure:
The system will grow until no further energy improvement is possible.
The primary objective is to maximize energy density
defined as $ρ_E = Total Bond Energy / Volume$
Since bond energy is constant, this is equivalent to maximizing bond density
$ρ_B = Number of Bonds / Volume$
The simulation therefore searches for the structure with the highest achievable bond density.
During the simulation the following quantities will be measured:
energy density
The simulation may produce several classes of structures:
The key result will be the maximum achievable bond density for the given geometric rules.
Possible later improvements include:
This simulation explores possible nucleus-like growth states as a stochastic graph-growth process. A structure starts from a central node and grows step by step along a fixed set of allowed spatial directions. Each node can use up to 12 resonance directions, and each new connection either creates a new node or merges into an already nearby node. In this way, the simulation does not grow arbitrary point clouds, but constrained resonance networks.
At every step, the simulation collects all possible growth candidates from all existing nodes and unused directions. Each candidate is scored according to three local criteria:
The next growth step is then chosen probabilistically, weighted by these scores. This means the simulation is neither fully deterministic nor fully random: it has a preferred direction toward compact, highly connected, resonance-compatible structures, while still exploring many alternative growth histories.
If a proposed new point lies close to an existing active node, it is merged and an edge is formed instead of creating a duplicate node. This allows loops, closed structures, and compact clusters to emerge naturally. Each accepted intermediate structure is evaluated and stored in an archive, grouped by a derived quantity (Q) and by the number of higher-connected nodes (k_P).
In addition to normal growth, the simulation occasionally explores a completed side-path. This side-path makes a deep copy of the current structure and then applies three operations:
This creates a more “finished” version of the current state without necessarily continuing the main growth path. If the completed side-path scores better, it can replace the current structure. This gives the simulation a useful shortcut: it can jump from a raw growing state toward a more complete resonance configuration.
Each structure is evaluated by a weighted scoring system. The score combines several geometric and topological criteria:
The total score is therefore not a physical energy yet, but a heuristic fitness measure. It favors compact, symmetric, well-connected, resonance-closed structures.
The current simulation can be understood as a resonance-guided growth model. It asks: if nucleus-like structures grow by occupying local directional channels, which configurations appear naturally when compactness, closure, symmetry, and high channel usage are rewarded?
The archive does not store only one best solution. Instead, it keeps multiple high-scoring variants per ((Q, k_P)) class. This is useful because different structural families can be compared at similar effective “mass” or complexity levels.
The approach is strong because it combines exploration and selection. It does not impose a final nucleus geometry directly, but lets candidate structures emerge from local rules. The use of fixed resonance directions also makes the model more structured than ordinary random graph growth. The archive system is especially useful because it preserves competing structural variants instead of collapsing everything into a single winner too early.
The completion side-path is also valuable. It mimics the idea that partially grown structures may later relax, close open channels, and discard weak extensions.
The scoring system is still heuristic. It rewards plausible structural properties, but it is not yet calibrated against measured nuclear data such as isotope stability, binding energy, decay modes, or known shell closures.
The growth process is also path-dependent. Early random decisions can strongly influence later structures, and the simulation may miss configurations that require temporarily unfavorable intermediate states.
Another limitation is that all edges are treated rather uniformly. Real nucleus growth would likely require different interaction types, such as proton-like, neutron-like, alpha-like, open-channel, closed-channel, and shell-specific contributions.
The current $Q$ measure is useful as an internal classification, but it is not yet physically fixed. It partly depends on connection degree and directional alignment, so its relation to actual nucleon number, charge, or isotope identity still needs calibration.
Future versions could improve physical realism by introducing explicit energy terms instead of pure geometry scores. For example:
The simulation could also separate growth from relaxation more clearly. A future version might first grow many raw candidates, then run a relaxation/optimization phase, and only then evaluate the final structures.
Another important improvement would be to track structural ancestry. This would show not only which final structures score well, but also which growth paths reliably produce them.
This version of the simulation is best understood as a resonance-topology explorer. It does not yet simulate real nuclei directly, but it searches for compact, symmetric, resonance-closed graph structures that may serve as candidate geometries for later physical interpretation. Its main strength is manual structural discovery; its next major step is calibration against real nuclear observables.
Use the share button below if you liked it.
It makes me smile, when I see it.