Area formula for growth forms of Tile(1,b) tilings

Prove that, for the aperiodic tilings generated by Tile(1,b), the area of the regular hexagonal growth form equals 2√3 times the area of the prototile Tile(1,b).

Background

Based on computational experiments and structural connections to the hexagonal lattice, the authors propose a specific quantitative relation between the prototile area and the growth form area for the Tile(1,b) family.

Establishing this formula would provide a concrete geometric invariant linking substitution rules to macroscopic growth-form geometry.