Hilbert Geometry on Polytopes: Volume Entropy and Simplex Bi-Lipschitz Equivalence — E8 Intelligence Research

FINDING: Hilbert geometry on convex polytopes is characterized by volume growth of metric balls and bi-Lipschitz equivalence to simplex geometry, linking hyperbolic tiling entropy to polytope structure. | MATH: Hilbert metric on polytope P: d_H(x,y) = (1/2) log( [a,x,y,b] cross-ratio ), where a,b are boundary intersections of line xy with P. Volume entropy h = lim_{R→∞} (1/R) log Vol(B(x,R)). For polytopes, h equals the entropy of the associated simplex geometry (bi-Lipschitz invariant). Hyperbolic tiling entropy: for regular {p,q} tiling, area of fundamental domain = π(1 - 2/p - 2/q) (Gauss-Bonnet), and growth rate of tiles = e^{hR} with h = √( (p-2)(q-2) - 4 ) / (2√(pq)) for the Poincaré disk (exact for {p,q} with 1/p + 1/q < 1/2). | CONNECTION: The cross-ratio in Hilbert metric is the same projective invariant underlying the golden ratio: for a segment [0,1], cross-ratio [0, x, 1, ∞] = x/(1-x). Setting x = 1/φ = 0.618 gives cross-ratio = φ = 1.618. The volume entropy of hyperbolic t Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com