The Violence of Equilibrium
A simplified model of territorial formation
I became interested in the evolution of national borders while working on my latest project for the Ars Electronica residency. I kept thinking about how these lines on a paper are really traces of internal and external pressures, constantly pushing territories to expand, contract, fracture, or disappear.
I wanted to see whether that movement could emerge from a very simple computational model. A model where pressure becomes a topological force, slowly shaping the outline of a territory.
This experiment is part of a broader research into mapmaking as a way of depicting desire. The full project link is at the end.
After a series of iterations, I settled on a simple diffusion algorithm. Each cell stores a faction and a pressure value, which is continuously exchanged with neighboring cells of the same faction. At the border, this pressure no longer circulates freely. It becomes a force acting against the resistance of another territory. When that force grows strong enough, the cell changes faction and the border moves.
In other words, once a territory is created, it expands until its internal pressure relaxes. It moves most easily into areas of lower pressure, whether these are neutral zones or weaker neighboring territories.
I chose a hexagonal grid because each cell has six neighbors at the same distance. This creates a more balanced field than a rectangular grid, where cells connect more strongly along horizontal and vertical axes. Pressure can therefore spread in more directions with fewer visible directional biases, allowing borders to curve, branch, and settle in a way that feels less imposed by the geometry of the grid.
The system is computed through custom CUDA kernels. Every cell reads its neighbors and updates at the same time, allowing pressure to diffuse and borders to shift across thousands of cells in real time. This parallel structure also mattered conceptually, because the territory is expressed as a field of pressure. Its shape is not decided from above. It emerges from many small local interactions happening at once.
The model can be reduced to a few simple rules.
- Every cell belongs to a faction and stores a pressure value.
- Pressure spreads between neighboring cells of the same faction, moving from higher pressure toward lower pressure.
- When a cell touches another faction, pressure becomes a competing force rather than simple diffusion. The neighboring faction with the strongest pressure makes the capture attempt.
- If that pressure overcomes the resistance of the cell, the cell changes faction. Its new pressure then allows the front to continue moving.
So the territory is never moved directly. Only pressure is added. The border is the visible result of how that pressure redistributes and competes locally.
Pressure from the same faction is allowed to circulate, so the cell gains or loses a small amount depending on the difference with its neighbors. All of those exchanges are combined into one value, the diffusion delta, which tells us how the local pressure changes during that step.
Once all six neighbors have been checked, the cell updates its pressure. Then it compares the strongest foreign offer against its local resistance. If that offer is strong enough, the cell changes faction. This same cycle is repeated across the whole grid, so the border keeps adjusting one local interaction at a time.
The result begins to resemble a density field. Pressure radiates outward from each faction, strongest near the points where it is introduced and gradually relaxing as it spreads through the territory. Where these fields meet, the gradients begin to shape the border.
In that sense, the system starts to echo other boundaries found in nature, such as competing bacterial colonies, where a visible edge appears from the interaction between two growing fields rather than from a line drawn in advance.
Once people can choose where to place a new faction, or add pressure to an existing one, the model becomes a form of gameplay. The choice is simple: claim territory for yourself or strengthen someone else. Their decisions enter the field as pressure and gradually reshape the borders around them.
What interests me here is how such a limited mechanic can start to reveal something about the individual. Faced with the same map, one person may expand their own claim while another chooses to reinforce someone else. The interaction becomes less about winning and more about making visible the kind of territorial instinct each person brings into the system.
In the next article, I will show how I used this pressure system to generate the structure of a map. The simulation defines the territories and their borders, which then become the basis for the visual landscape.
LINK - NEXT ARTICLE - TBA
LINK - PROJECT - TBA
