From Right Triangles to Fibonacci-Type Spirals via the Hypotenuse-Axis Intercept (HAI): Seed Ratios, Möbius Maps, and the Golden 2:1 Triangle

This paper introduces the Hypotenuse–Axis Intercept (HAI) procedure (Section 2), a geometric mechanism that generates Fibonacci-type spirals from a single right triangle. One HAI step fixes a rectangular seed (S₁, S₂); thereafter the spans satisfy the universal recurrence Sₙ₊₁ = Sₙ + Sₙ₋₁ (n ≥ 2) (Proposition 3.3), with radii Rₙ = Sₙ for n ≥ 2 (Definition 3.6). Key points Golden seed. S₂/S₁ equals the golden ratio φ = (1 + √5)/2 iff the triangle’s legs are in the ratio 2:1 (Theorem 4.5). Pythagorean triples. The four canonical HAI seed ratios form the positive orbit of t = p/q under a dihedral subgroup of PGL₂(ℚ) (Propositions 5.3 and 5.7). Degenerate limit. The classical 1:1 Fibonacci seed appears as a degenerate HAI limit (Remark 2.6). Interactive supplement GeoGebra simulation: https://www.geogebra.org/calculator/cbgzabjc