Substitution tilings build infinite aperiodic structures through repeated inflation — replace each tile by a cluster of smaller tiles according to a fixed rule, then rescale. Penrose tilings are the most famous example: two rhombus shapes, a substitution rule, and an inflation factor of the golden ratio phi. The resulting tiling has 10-fold rotational symmetry but no translational period. It's ordered without repeating.
Imura constructs a substitution rule for 10-fold symmetric rhomb tilings with inflation factor phi cubed — a larger inflation step that replaces each tile with a more complex cluster — and proves it is recognizable. Recognizability means the substitution can be uniquely reversed: given the tiling, you can determine which tiles were grouped together at the previous hierarchical level, and at every level above that, without ambiguity. The hierarchical structure of the tiling is not hidden. It's recoverable from local information alone.
This is not automatic. Many substitution rules are not recognizable — they produce tilings where distinct hierarchical decompositions are locally indistinguishable, so no finite patch determines where you sit in the hierarchy. Recognizability requires that the substitution rule imprints enough local information at each level to distinguish itself from all other possible decompositions.
The proof works by showing that every legal vertex configuration in the tiling uniquely determines its parent tile in the next hierarchical level. The determination is local — it depends only on the immediate neighborhood of the vertex, not on the global structure of the tiling. Since each level determines the next, the entire infinite hierarchy is recoverable by iterating the local recognition.
In the right kind of ordered system, local information contains global structure. The hierarchy is recognizable because it left fingerprints at every scale — and each fingerprint is small enough to read from where you stand.