Scale-Optimized Geometric Acceleration of Polygonal Approximations to π

This paper develops a scale-bb Richardson–Lagrange acceleration framework for polygonal approximations to π\\pi. Starting from the half-perimeter PN=Nsin⁡(π/N)P_N=N\\sin(\\pi/N) of the regular 2N2N-gon inscribed in the unit circle, the method combines values at geometrically scaled resolutions N,bN,b2N,…N,bN,b^2N,\\ldots using Lagrange extrapolation weights chosen to cancel successive even powers of N−1N^{-1}. The resulting order-rr accelerated approximation achieves an error of order O(N−2r)O(N^{-2r}). The paper establishes explicit weight formulas, a uniform stability bound for geometric scale nodes, and analytic remainder estimates. High-precision numerical verification confirms the predicted convergence orders. The constructibility analysis distinguishes exact dyadic refinement, obtainable through repeated square-root constructions, from tripling-based refinement, which generally requires solving a cubic and is therefore not compass-and-straightedge constructible in general. Approximate tripling architectures are treated separately with explicit perturbation control.