Tilt angle of the hat tiling’s growth hexagon

Prove that the regular hexagonal growth form of the hat tiling is oriented with tilt angle γ equal to −α, where tan(α) = √3/(3 + 2τ) and τ is the golden ratio, yielding γ ≈ −0.270919.

Background

The hat tiling’s growth form is established to be a regular hexagon, but its precise orientation relative to the underlying lattice is unknown. Using triangulation line slopes derived from structural properties, the authors propose a specific tilt angle.

Confirming this conjecture would complete the geometric description of the hat tiling’s growth form.