Cross-Ratio as Projective Invariant Linking Harmonic Ranges to Hilbert Geometry — E8 Intelligence Research
Year: 2026
Type: preprint
Abstract: FINDING: The cross-ratio is the fundamental projective invariant, and its special case (harmonic range = −1) connects directly to Hilbert geometry of convex polytopes, which encodes metric structures via projective transformations. | MATH: Cross-ratio of four collinear points: (A,B;C,D) = [(C−A)(D−B)]/[(C−B)(D−A)]. Harmonic range: (A,B;C,D) = −1. In rational trigonometry, this becomes a purely algebraic invariant using squared distances (quadrance) and spreads, avoiding transcendental functions. Hilbert metric on a convex domain Ω: d_H(x,y) = (1/2)·|ln[(A,B;x,y)]| where A,B are the intersection points of line xy with ∂Ω. For a simplex, this yields a normed space isometric to (R^n, ||·||_∞). | CONNECTION: The harmonic range (cross-ratio = −1) is the projective analogue of the golden ratio's self-similarity — it is the unique non-trivial involution on a line. The Hilbert metric's volume growth for polytopes relates to the Mahler volume and to root systems (A_n, B_n, C_n, D_n) via the sim Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
Author: Caldin, Andrew Stewart
Language: English
Topic: Mathematics and Applications
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7204136498
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