Icosahedral Bond Network Simulation

Purpose

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:

• directional bonding
• geometric compatibility
• frustration of the grid
• bond density per volume

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.

Discussions and RQM correlations

Overview of the Icosahedral Bond Network Discussion

1. What We Actually Formulated

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

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:

• 12 rays corresponding to the vertex directions of anicosahedron.

Bonds

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.

Growth Rule

The network grows by:

• adding bonds along allowed ray directions
• connecting nodes when neighbors exist in the grid

Objective

Maximize energy density, which is equivalent to maximizing: bond density per volume because each bond carries the same energy.

Frustration

Because icosahedral symmetry cannot tile Euclidean space perfectly, the network inevitably develops:

• incomplete coordination
• angular mismatch
• defects

The simulation aims to quantify these effects.

Symmetry

The system we aim for should also have these constraints:

• Small momentum bias in respect to rotation
• An optimal symmetry along at least on axis, preventing wobbling and angular momentum
• This is required for stable orbital electrons
• Not only vector balance (first moment = zero) but also second moment isotropy –> Rotationally isotropic to second order

Which geometrical figures allwo this:

• 4 directional tetrahedron
• 6 directional octahedron
• 12 directional icosahedron

Even if we do not force any extra penalties, this will produce:

• no preferred axis
• nearly spherical energy density if the lattice allows it.

So by using icosahedral structure might automatically optimize for symmetry if we optimize for density

Energy 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.

2. Key Insights from the Discussion

Insight 1 – Icosahedral geometry naturally produces 12 preferred directions

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.

Insight 2 – Icosahedral symmetry cannot form a periodic lattice in 3‑D

Five-fold symmetry conflicts with translational symmetry.

Result:

• global tiling impossible
• local order possible

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.

Insight 3 – The system resembles a valence network

Each node has:

• 12 directional bonding slots

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.

Insight 4 – The optimization problem splits naturally

Two layers appear:

Topological problem

Which bonds exist?

Geometric problem

How much distortion is needed to embed the network in real space?

Separating these simplifies simulation design.

Insight 5 – The objective function is unusual

Most physical simulations minimize total energy.

This model maximizes:

bond density per volume

That emphasizes packing efficiency of directional bonds.

3. Relation to Standard Physics

Several components already exist in known physics.

Icosahedral Cluster Physics

Metal clusters often adopt icosahedral structures.

Examples include Mackay clusters with magic numbers:

13, 55, 147, 309

These correspond to successive shells.

Geometrical Frustration

A major topic in condensed matter physics.

Occurs when: local order ≠ global tiling

Examples:

• spin ice
• curved crystals
• quasicrystals

Patchy Particle Models

Particles with directional bonding sites used in simulations of:

• self-assembly
• network gels
• quasicrystals

The rays we are going to simulate here correspond closely to patch directions.

Valence Network Models

Used in systems such as:

• network glasses
• covalent crystals
• polymer networks

Focus is on:

• bond topology
• bond angles

rather than pair potentials.

4. Where the Model Becomes Less Standard

12‑valent directional nodes

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.

Optimization for bond density

Most models optimize:

• energy per particle
• free energy

Here the focus is: bonds per volume which is closer to a packing efficiency problem.

Predefined directional grid

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:

• Rotonal resonance channels are based on distance-locking, so the bonds are preferably multiples of a given length.

5. Closest Research Directions

Geometrically Frustrated Assemblies

Key concept: local order incompatible with global space

Important authors include:

• Irvine
• Vitelli
• Chaikin

Patchy Particle Self‑Assembly

Particles with directional bonding sites used to model:

• protein assembly
• colloidal self‑assembly
• quasicrystals

Important researchers:

• Francesco Sciortino
• Daan Frenkel

Icosahedral Cluster Physics

Classic work by:

• Jonathan Doye
• David Wales

Focus areas:

• cluster growth
• structural transitions
• geometric frustration

Quasicrystal Physics

Icosahedral symmetry appears naturally in quasicrystals.

Key concept: long‑range order without periodicity

Important figures:

• Dan Shechtman
• Paul Steinhardt

Network Glass Physics

Valence networks studied in materials like:

• silica
• chalcogenide glasses

Important theory: Phillips–Thorpe rigidity theory.

Further Mathematical Connection

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.

6. What the Simulation Might Reveal

The key question becomes:

How much bond density can an icosahedral directional network sustain before frustration forces defects?

Possible outcomes include:

1. compact clusters
2. defect‑mediated extended networks
3. quasiperiodic structures
4. amorphous high‑density networks

7. Core Conceptual Question

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:

• geometry
• condensed matter physics
• network theory
• quasicrystal physics

Base Geometry

Nodes

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

Rays correspond to the 12 vertex directions of an icosahedron.

Properties:

• originate at a node
• point toward potential neighbor nodes
• define allowed bond orientations

Two opposite rays form a ray axis.

6 axes
12 rays

Bonds

A bond is formed when two nodes connect along opposite rays.

node A —- bond —- node B

Properties:

• finite length
• fixed direction defined by rays
• connects exactly two nodes

Not all rays necessarily lead to valid neighbors in the grid.

Therefore nodes may have:

• 12 bonds (rare)
• 6 bonds (common)
• 3 bonds (outer nodes)

Energy Model

Energy is stored almost entirely in bonds.

Bond Energy

Each bond contributes

Ep / 3

to the total energy.

Node Energy

Nodes may carry a very small intrinsic energy

p < 0.1% Ep

This contribution can be neglected for the initial analysis.

Frustration

Because icosahedral symmetry cannot tile Euclidean space perfectly, the grid will contain frustration.

Frustration appears as:

• angular mismatch at nodes
• incomplete bonding
• missing neighbors
• distorted bond propagation

The simulation will track these defects explicitly.

Simulation Strategy

The network will be constructed bond by bond.

Procedure:

1. start from a seed node
2. extend the structure along available rays
3. create bonds whenever compatible neighbors exist
4. monitor bond density and frustration

The system will grow until no further energy improvement is possible.

Objective

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.

Observables

During the simulation the following quantities will be measured:

Structural

• number of nodes
• number of bonds
• average coordination number
• fraction of occupied rays

Geometric

• bond orientation deviations
• angular mismatch at nodes
• distribution of coordination numbers

Topological

• number of frustrated nodes
• number of dangling rays
• defect structures

Thermodynamic proxy

energy density

Expected Outcomes

The simulation may produce several classes of structures:

1. compact clusters with minimal frustration
2. extended networks with distributed angular strain
3. defect-mediated structures
4. possibly quasiperiodic arrangements

The key result will be the maximum achievable bond density for the given geometric rules.

Future Extensions

Possible later improvements include:

• angular strain energy
• bond length elasticity
• rotational freedom of nodes
• stochastic growth processes
• Monte Carlo optimization