Redistricting Compactness as Constrained Perimeter Minimization: Soap Bubble Theory and Discrete Approximation

We propose a mathematical framework for redistricting compactness grounded in the classical soap bubble problem: the partition of a planar region into N subregions of prescribed population that minimizes total shared boundary length. This formulation, which we call the soap bubble model, yields a physically and mathematically natural optimality criterion that generalizes all existing per-district compactness measures. We carefully distinguish two formulations: minimizing physical boundary length in geographic coordinates, and minimizing a density-weighted length in the coordinates produced by the Monge–Ampere density-flattening map. In the transformed coordinates, the first-order conditions for a minimizer are precisely the classical two-dimensional Plateau’s laws – circular arc boundaries, 120-degree triple junctions, and perpendicular boundary contact. In geographic coordinates, the Euler–Lagrange equation shows arc curvature varies proportionally with local population density, so boundaries curve more tightly through dense areas and gently through sparse ones. Remarkably, the 120-degree junction condition and 90-degree boundary condition are unchanged by the density, as they follow from a force-balance argument that does not involve density. We adopt the physical-length objective as the natural redistricting criterion – it minimizes actual miles of district boundary and requires no coordinate transformation – and derive the corresponding density-modified volume-preserving mean curvature flow as its gradient descent. The N-state Potts model on the census block adjacency graph with population-weighted pressure updates is the natural discrete implementation of this flow. We define a multiplicative gerrymandering metric decomposed into tactical and systemic components and discuss practical computation, including grid-pinning effects in the discrete setting.