It’s a principle I work with when cutting out silhouettes. Paper is fragile, particularly when cut this fine, and although it’s light, it still has enough weight to tear itself.
There are variations and opinions on the the idea of the “rule of three”. But tradition and culture and habit aside, it’s in the editing checklist less for its fairy-tale echoes and more for its properties as a physical structural corrective to floppy story elements.
One of the few practical constructions skills I remember from helping (helping?) my dad around the property (apart from the fact a Cobb & Co hitch is often one of the strongest elements of a building), is the importance of a brace — the beam or pole or cross-limb that creates a rigid triangle and stops objects leaning slowly sideways. Think of the planks that make the diagonals of the “Z” on the stereotypical barn door.
That inherent structural stability of triangles is the reason that finally made the idea of three references or repetitions of a clue, background element, etc, make sense to me.
One reference (or hint, or point of contact) can be fine, if the material is sturdy enough — which paper isn’t. Lots of references can be stylistically fun, if not unwieldy — in a silhouette, they can create confusion, until you only have a field of light and shadow with no sense to it.
But in case of doubt, three little anchor points can be enough to create a stable field within the story, and enough of those form a spiderweb that can hold together quite a fragile lace.