Geometry of hexagonal growth forms in hat and Tile(1,b) tilings

Determine the edge lengths and the orientation (tilt angle) of the regular hexagonal growth forms associated with the hat tiling and its deformed variants Tile(1,b) introduced by Smith et al., including explicit numerical values or formulas.

Background

The hat tiling and its Tile(1,b) deformations are aperiodic substitution tilings whose growth forms are deduced to be regular hexagons. Despite this structural understanding, their precise geometric parameters (edge lengths and tilt angles) have not been determined.

The authors state this gap explicitly and seek exact characterization of these hexagonal growth forms’ geometry.