When each cell adopts the opinion of its majority, domains form, boundaries straighten, and one opinion wins.
A town where each person copies the majority of their neighbors each day, with no persuasion or media or leaders.
Within a few steps, random noise crystallizes into domains separated by sharp curved borders. Borders straighten, small domains shrink, and one opinion takes the whole grid — the same dynamics as merging soap bubbles.
Each cell counts colors in its Moore neighborhood (8 surrounding + self = 9 votes, no ties) and adopts the majority.
The first few steps snap noise into a map of large regions with curved borders. After that, borders drift to reduce curvature, small pockets get eaten, and the grid settles on a single color.
Figure 1. Majority rule on a 200x200 grid. Switch between the 9-cell Moore neighborhood (self + 8 surrounding) and the 5-cell von Neumann neighborhood (self + 4 orthogonal). Both have odd group sizes so ties never arise. The smaller group produces noisier boundaries and traps the dynamics in frozen stripes more often. Playback pauses automatically once the grid stops changing.
A cell deep inside a single-color patch never changes. A cell on a curved border gets outnumbered on the concave side and flips. Curved borders retreat; small regions collapse fast.
Straight borders are stable — every border cell sees roughly equal votes. The late-stage dynamics are slow because surviving boundaries are nearly straight.
Domain size $L$ scales as the inverse square root of boundary density. Theory predicts $L \propto \sqrt{t}$ — the Allen-Cahn scaling law from phase ordering.
Figure 2. Domain coarsening. The chart tracks domain size over time, with the dashed blue line showing the theoretical \( \sqrt{t} \) scaling. The fitted exponent is a least-squares slope of \( \log L \) against \( \log t \) over the recent history; it should hover near 0.5 while coarsening is active and drop once the system locks into frozen stripes or consensus. Boundary cells are highlighted in blue when toggled on.
Each boundary segment moves at a velocity proportional to its local curvature: convex bumps retreat, concave dents fill in. The same curvature-driven motion governs grain growth in cooling alloys, magnetic domain ordering, and foam evolution.
With 50/50, the winner is random. Whichever color forms larger early domains gets a compounding structural advantage. Symmetry breaks: nothing in the rule favors either color, yet one always wins.
Add a slight bias — 52% black — and black always wins, but slowly. At every boundary, the black side has marginally more reinforcement, so black regions expand on average.
Figure 3. Bias and nucleation. Adjust the initial fraction of black cells and watch the dynamics change. "Run sweep" tests each bias from 50% to 60% and plots the consensus time.
The same Allen-Cahn equation governs phase separation in alloys, magnetic domain evolution, and grain boundary formation. Length scales as $\sqrt{t}$, boundaries move with their curvature, and late-stage morphology is statistically self-similar.
The majority rule is one of the simplest opinion-formation models. The winner isn't determined by argument strength but by the geometry of initial fluctuations.