3D Twenty-Vertex Model Simulation vertex-models

Alexey Bufetov and Panagiotis Zografos (definition), Leonid Petrov (implementation)

Simulation Info

3D Twenty-Vertex Model Simulation vertex-models

Alexey Bufetov and Panagiotis Zografos (definition), Leonid Petrov (implementation)

About this simulation

This simulation demonstrates the twenty-vertex model introduced by Bufetov and Zografos (work in progress). The model consists of a 3D lattice where arrows are placed on edges, always pointing in positive coordinate directions. Each vertex has 8 possible incoming arrow configurations and 8 possible outgoing configurations, with conservation of arrow count through each vertex.

The model has 20 vertex types: 2 deterministic (000→000 and 111→111) and 18 stochastic configurations with tuneable weights. The sampling proceeds in time slices where t = x + y + z, with boundary conditions: empty in xz and yz planes, full in xy plane.

Model Parameters

Vertex Weights (12 Free Parameters)

100→100:

100→010:

010→100:

010→010:

001→100:

001→010:

110→110:

110→101:

101→110:

101→101:

011→110:

011→101:

Note: These 12 parameters, along with sum-to-one constraints, determine all 18 stochastic vertex weights. The notation abc→def means incoming arrows from directions (x,y,z) and outgoing to (x,y,z), where 1=arrow present.

Visualization Controls

X-direction

Y-direction

Z-direction

Orthographic Density Projections

Arrow directions mapped to RGB channels: X→Red, Y→Green, Z→Blue. Shows flow density accumulation when collapsing one spatial dimension.

XY Plane (sum over Z)

XZ Plane (sum over Y)

YZ Plane (sum over X)

Time-Slice Activity

Arrow counts by direction over time slices t = x + y + z. Shows temporal dynamics and growth fronts.

Height Map from Cube Occupancy

Grayscale height map H(x,y) = Σ_z filled_cubes(x,y,z) with optional contour lines showing bulk geometry structure.

Cross-Section Slicers

View 2D cross-sections of the 3D vertex model at constant coordinate planes. Edge occupancies shown as colored line segments: Red (X-direction), Green (Y-direction), Blue (Z-direction). Only edges lying within the slice plane are displayed.

X = 5 (YZ plane)

Y = 5 (XZ plane)

Z = 5 (XY plane)

code

(note: parameters in the code might differ from the ones in simulation results below)

Link to code (This simulation is interactive, written in JavaScript, see the source code of this page at the link)
Link to code (C++ code for the simulation)

Dear colleagues:

Feel free to use code (unless otherwise specified next to the corresponding link), data, and visualizations to illustrate your research in talks and papers, with attribution (CC BY-SA 4.0). Some images are available in very high resolution upon request. I can also produce other simulations upon request - email me at lenia.petrov@gmail.com