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

The Hypotenuse–Axis Intercept (HAI) ignites a Fibonacci-type construction from a single right triangle. Rotating the hypotenuse about an acute vertex until its endpoint meets the extension of the adjacent leg axis produces a positive seed (S1,S2)(S_1,S_2)(S1,S2) on perpendicular axes (four canonical branches). The seed is metrically complete: a characteristic HAI identity linearises the Pythagorean relation and yields explicit inverse formulas recovering the generating right triangle from (S1,S2)(S_1,S_2)(S1,S2) once the branch is fixed. Adjoining squares then forces a rectangle cascade with Sn+1=Sn+Sn−1S_{n+1}=S_n+S_{n-1}Sn+1=Sn+Sn−1, and quarter-circle arcs of radii Rn=SnR_n=S_nRn=Sn for n≥2n\ge2n≥2 form a Fibonacci-type spiral. The triangle enters the subsequent dynamics only through the initial ratio ρ2=S2/S1\rho_2=S_2/S_1ρ2=S2/S1; thereafter ρn+1=1+1/ρn\rho_{n+1}=1+1/\rho_nρn+1=1+1/ρn is a universal Möbius update represented by the Fibonacci transfer matrix M∈PGL2(Z)M\in \mathrm{PGL}_2(\mathbb Z)M∈PGL2(Z), driving ρn→φ\rho_n\to\varphiρn→φ with sharp error Θ(φ−2n)\Theta(\varphi^{-2n})Θ(φ−2n) unless the seed is golden. We record the classical trace identity tr(Mn)=Ln\mathrm{tr}(M^n)=L_ntr(Mn)=Ln, linking Lucas numbers to the ratio evolution. A complex lift gives centre-step vectors with a fixed quarter-turn and asymptotic scale λk→φ\lambda_k\to\varphiλk→φ; this becomes an exact one-step similitude precisely for the golden seed. We prove that ρ2=φ\rho_2=\varphiρ2=φ for some canonical branch if and only if the leg ratio is 2:12{:}12:1, i.e. (a,b,c)=(r,2r,r5)(a,b,c)=(r,2r,r\sqrt5)(a,b,c)=(r,2r,r5). Finally, the four canonical ignition ratios form the positive part of a dihedral D4D_4D4 orbit of the half-angle parameter t=cot⁡(β/2)t=\cot(\beta/2)t=cot(β/2) under Möbius maps in PGL2(R)\mathrm{PGL}_2(\mathbb R)PGL2(R) (restricting to PGL2(Q)\mathrm{PGL}_2(\mathbb Q)PGL2(Q) when t∈Qt\in\mathbb Qt∈Q). A tangent-based recovery law for arc centres applies already at ignition, and we also note a trirectangular tetrahedron companion with orthogonal edges (r,r,2r)(r,r,2r)(r,r,2r). How to cite this version: Douglas Larsson Engholm. "From Right Triangles to Fibonacci-Type Spirals via the Hypotenuse-Axis Intercept (HAI)". Version 4.0.0, Zenodo (2026). DOI: 10.5281/zenodo.18401871 BibTeX for researchers: @misc{LarssonEngholm_HAI_2026_v4, author = {Larsson Engholm, Douglas}, title = {From Right Triangles to Fibonacci-Type Spirals via the Hypotenuse-Axis Intercept (HAI): The Characteristic Identity, Möbius Maps, Lucas Traces, and the Golden 2:1 Triangle}, year = {2026}, month = feb, publisher = {Zenodo}, version = {4.0.0}, doi = {10.5281/zenodo.18401871}, url = {https://doi.org/10.5281/zenodo.18401871}}