Isoperimetric Flow: Why the Circle Wins

Of every closed loop with a given perimeter, the circle is the only one that maximizes the area it encloses — a result the calculus of variations proves by showing that a curve of constant curvature is the unique stationary point of the area functional under a fixed-length constraint. This simulator makes that proof visible: a closed polygon of adjustable resolution evolves under the exact gradient flow implied by the Euler–Lagrange condition, each vertex sliding along its outward normal by an amount proportional to how far its local curvature departs from the circle's constant curvature, while the total perimeter is held fixed at every step. Watch a jagged star, a blob, a square or an ellipse relax into a near-perfect circle as the isoperimetric ratio A/L² climbs toward its theoretical ceiling of 1/4π, with live readouts of perimeter, area, the curvature deviation that drives the flow, and a real convergence chart of the measured ratio against the bound.