Decide existence of growth form for substitution tilings

Establish a universal decision procedure to determine whether a given substitution tiling possesses a growth form, in the sense that the scaled coordination shells converge (in Hausdorff distance) to a bounded (d−1)-dimensional surface independent of the initial patch.

Background

Substitution tilings are generated by inflating prototiles via a linear map and dissecting them according to fixed rules, producing nonperiodic structures of great interest. While periodic and regular grid tilings have established methods for computing growth forms, the existence and determination of growth forms for substitution tilings is largely unresolved.

The authors emphasize this as a central challenge, seeking a universal decision procedure to determine whether a substitution tiling admits a growth form under the coordination-shell/graph-distance notion used throughout the paper.