Dynamic Foam Emerging from a Gas of Vibrating Particles

 The hypothesis is that the Dynamic Foam ...

... can emerge out of a gas of particles that have an attraction-repulsion force that's in-sync with their vibrational rhythm. 

The particles are spherical solitons going back and forth in size: 1 <---> 10

Their colour changes depending on their state:  Red (1) <---> Yellow (10)

They repel their neighbours while expanding (+) vs attract when contracting (-)

• Two neighbours with different timing can be contracting and attracting each other, but when one starts to expand again before the other, it will push that neighbour away -> chaos

• Two neighbours with the same timing are in harmony and bounce together -> spherical or linear group

• Random --> Voronoi

Linear formation and pulsing edges.

Spherical groups (credit to Rawdy):

The initial condition of the gas are particles with a random timing where some are expanding while others are contracting:

Think of this distribution model in Unity: 

https://github.com/przemyslawzaworski/Unity-Million-Spheres

https://x.com/pzaworski90/status/1641518112087367680 (Clip)

The idea is similar to these Spin Wheels: 

https://www.complexity-explorables.org/explorables/spin-wheels/

That is related to the Kuramoto model:

https://en.wikipedia.org/wiki/Kuramoto_model

https://youtu.be/FO3mYe8zkrM

But those particles synchronise, losing their fixed period, like metronomes placed on a rolling table:

https://en.wikipedia.org/wiki/File:Wesphysdemo_-_Synchronized_Metronomes.webm

While the ‘unique' thing of my model is that all the particles keep their fixed period, and as a result we 'might' get pulses in the edges of the Voronoi mesh.

Mockups in Midjourney

So long story short I haven't been able to make any significant leap forward to boost up the simulator to a large scale, so I've switched now to make an animation of the whole concept and made some mockups in Midjourney. To be continued ...

Physics of the Dynamic Foam

• In the beginning Space was a dense medium wherein vibrations appeared (Black -> white).

• The tiny vibrations are like gas-particles and Space became a dense misty cloud.

• At random points in this cloud the vibrations started to align and harmonise (blue).

• The expanding harmonious dots collided and form a pressure regions (red).

• A Voronoi pattern (foam) formed and the edges distributed the intense pressure.

• The pressure distribution can’t make <90°-turns and is cut-off at certain junctions.

• The strength of the currents in the edges affect the size of the cells: heating up vs. cooling down. Changing cells-sizes change consequentially the angles … and the mesh becomes dynamic.

• Stable fluctuations emerge that form strings that can turn into knots.

• Gradual pathways 'pinch' Space. 

It's like the idea of a Quantum Foam where particles pop into existence.

 Proposed by John Wheeler:

Flow regulated by Local Tree Network:

 The idea was to simplifying the Monte-Carlo-Marcov-Chain method to just a small Local Tree Network.

a. It’s still the same starting idea of junctions are open ( > 90°) or closed ( < 90°)

b. The simplification was to calculate the weight of each edge based on the number of connections with only small local percolation-tree, that should do:

c. The total weight of the edges of around a cell defines its pressure. 

d. The pressure changes between cells pushes them further or closer together.

Here’s a small test by Markus Rawdy who came up with this sim in Houdini:

The Fabric of Space

 The Fabric of Space is a Semi-Solid like a Foam .

Think of this Truchet FBM Lace toy by Fenix to get a feeling:

1. Currents in the edges.

2. When a junction is open current can pass.

3. Depending on the Force of the current a Bubble/Field can Expend or Shrink.

We can model this foam with a tri/tet mesh.

4. When there is a lot of current in ‘voronoi’ edge Y between A,B than the ‘delaunay’ edge between A-B contracts, otherwise it expands.

The diagram below shows the different parts of calculating the Gates, using a Graph Network to calculate the flow in all the Edges, and finally how the mesh contracts or expands at different parts.

Note, the idea is that by using a small local tree this whole Graph-Network is no longer necessary, see next Local Tree Network post. 

Processing Schemes & Diagrams II

 1. A tri/tet (Delaunay) mesh is the physical backbone.

2. Via the Barycenters we can check if ‘gates’ of it’s dual (virtual) Voronoi mesh are open or closed.

3. The results form a Graph Network.

4. With Monte Carlo Markov Chains (MCMC) walks we measure the currents.

5. The value of these currents defines the deformation of the dual tri/tet-edges.

6. Loop back to 2.

BTW when doing walks gates could be bridged and loops can emerge, therefore the need for 'Valves' that prevent U-turns in the Gates. Note, these diagrams are just a suggestion maybe there’s a better solution.

Little extra: Delaunay triangulation with Hilbert curve

Markov-Chains in VoroX

For VoroX.jl Benoît build a system that is similar to Google’s PageRank that uses Markov Chains, where web-crawlers are released onto the internet to measure the connections and generate rating of sites.

In his system there's an amount of traffic released into the Voronoi-edges of the foam, these pulses move from the edges to the junctions and check if the gates are open or closed. 

When a gate is open a pulse moves on to the next gate and so forth. These ‘ratings' correspond to the current in each edge between the cells and define if a cells shrinks or expands. The particles move until they reach a Steady-State, as in Nick’s 2D Dynamic Foam model:

If you’re not familiar with Markov-Chains, this video from Khan academy explain it in 3 minutes:

https://www.khanacademy.org/college-careers-more/bjc/2018-challenge/2018-challenge-mathematics/v/markov-chains-breakthrough-junior-challenge-2018

So in VoroX.jl dogs/pulses/web-crawlers move to other rooms/edges/websites when there’s an open-door/gate//link. 

After a couple of iterations a Steady-State is reached and we can count the dogs/flow/connections.

These values lets us know how much ‘erosion' the flow in an edge (a-c) has caused to its neighboring cells A and D, and as such the contraction-value for its dual edge A-D.

---

Details from the control panels of the 2D Dynamic-Foam

https://github.com/weigert/DynamicFoam

 that might make more sense now:

https://youtu.be/X-KytAnr1rg

https://youtu.be/wR3JgunXDdU

• Energy, Scale and Critically 

This refers to the number of particles/pulses are put into the system and their size.

 • Circumcenter / Barycenter / Incenter 

Refers to the method used to create the dual Voronoi Mesh on top of the Delaunay Mesh.

• Expansive Flow / Contractive Flow

Particles in Voronoi-edges (a-c) expand OR contract Delaunay-edges (AB).

Details from theVoroX.jl control panels of the 3D/2D Dynamic-Foam program where the particles are replaced by a Markov-Chain value system:

https://youtu.be/0UTzeL0uesY

https://youtu.be/sTa5utH-8Uc

Height: 

set size of the Area

Point size, Center size, Edge width, Circuit width: 

visualisation properties

Decay:

time a current in edges stays highlighted

Shading for the faces.

3D