Triadic Morley: A Size–Shape–Phase Decomposition of a Triangle

In this paper, I recast Morley’s construction as a triadic decomposition of triangle data into a scalar size Sz (e.g., the circumradius R), a similarity-class shape Sh (angles A,B,C), and a finite phase label Φ encoding trisector adjacency and orientation. Morley’s map is presented as a natural Size–Shape projection onto the 1D equilateral orbit with the explicit scale formula side(M(T)) = 8 R sin(A/3) sin(B/3) sin(C/3),and I formalize a generic local inverse (“Morley Lift”): given an oriented equilateral carrier E together with compatible Φ, the preimage triangle is locally unique up to orientation-preserving similarity outside a proper real-analytic exceptional set. A constructive Jacobian-based system and numerical certificate are included, along with complementary Morley-type couplings on the other two edges of the triad.