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.