The past months I have been in contact with MIT postdoc BenoĆ®t Legat a mathematician in numerical geometry, who has converted the 2D DynamicFoam model, developed by an other MIT affiliate Nick Weigert, into a 3D application:

**Voro-X**

The name stands for Voronoi diagrams (foam) where energetic currents run through the edges of the mesh, and X-ings (Crossroad / Junctions) that act like a gates where currents are switched OFF or ON based on the sharpness of the angles. (0/I)

The level of current passing through the edges, has river-like effects where 'sedimentation' makes the cells grow; and 'erosion' makes them shrink. The changes of the cell-size changes the angles, and thus the direction of the currents: currents can switch. This whole interaction model based on a few rules generates a CA type of Dynamic Foam.

Voro-X is build with the programming language Julia and you can download the program here:

http://www.800million.org/Vorox.zip

https://github.com/blegat/VoroX.jl

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__Voro-X compared to Nick's 2D DynamicFoam (see Wiki-page):__

- The criticality used is equivalent to an infinite criticality and cannot be changed.
- If "Edge scale" is on, the dynamics is equivalent to DynamicFoam: it scales the edges. If it is off, it scales delaunay simplices.
- The "Height" setting changes the height of the rendering frame but does not influence the dynamics. The points are are samples in a fixed square or cube that cannot be changed.
- The "Periodic" setting determines whether the meshes is considered infinitely periodic along each directions.
- The "Voro shading" determines whether the Voronoi cells are displayed (if it is on) or the Delaynay cells (if it is off). The transparency of the shading is determined by the "Transparency" setting.

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__Installation__

The program works on Windows, Mac and Linux,

but you first need to install Julia.

https://julialang.org/downloads/

Next dowload and unzip the Voro-x package/repository:

http://www.800million.org/Vorox.zip

Decompressed the package and rename it to for instance: Desktop/VoroX

Launch Julia and you get the prompt:

julia>

Type ] so that the prompts becomes:

(@v1.7) pkg>

(Note, backspace '<-' will bring back the julia> prompt )

Next activate VoroX.jl by typing:

(@v1.7) pkg> activate ~/Desktop/VoroX

(or via an other directory/name where you have placed the VoroX folder)

Now install its dependencies as follows:

(VoroX) pkg> instantiate

The installation will take a while.

Once this is done, to launch the software in Makie.jl, with K points and N dimensions (K, N), switch back to 'julia' by using 'backspace' and type:

julia> using VoroX

and finally the last command:

julia> foam(10, 3)

(This might take a longer time to load: 5 to 10 min.)

Note that N can only be 2 or 3 (for 2D or 3D)

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See screenshot for how it looks in the Terminal on my Mac,

all in all it are just 4 input lines in Julia:

(@v1.7) pkg> activate ~/Desktop/VoroX

(Vorox) pkg> instantiate

julia> using VoroX

julia> foam(20, 3)

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Next up is scaling this system up, to run with millions/billions of cells, get rid of some of the bugs, use optimum shading to visualise the foam and look for emergent pattern, knots!