Tweedle Dee - Tweedle Dum

 Random Vibrations vs Harmony

Pressure is Balanced Out vs Colliding

In this 3rd video there are as of 1:44 some different Rendering Modes:

Group Color : where each group-size of particle has it's color: White, Red & Blue

Single color: All particles in white

Period of the particles: Where they change color as they change in size (contract-attract/expand-push)

Density: mainly in Red

Size: Different tones of Yellow

The program is made in GODOT and will be posted soon on Git. Credit to David S. for the development and his foundation.

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