Can one collection of shapes be cut apart and reassembled into another? A question about scissors — that no algorithm can answer.
Given two finite collections of polygonal shapes in the plane:
A = {P1, P2, …, Pn}each with rational vertex coordinates. You are allowed to:
Can collection A be reassembled to form collection B?
It looks like a jigsaw puzzle. Something a child could play with.
But this question carries the full power of computation inside it.
Through reductions from known undecidable tiling problems (Berger, Robinson), shape dissection equivalence becomes undecidable.
There is no universal procedure that decides geometric reassemblability.
This is geometry carrying computation.
For any two specific collections, the answer is either yes or no.
But there is no single algorithm that works for all inputs. The pattern of which pairs are equivalent and which are not encodes an uncomputable function.
Scissors and glue — and yet, beyond the reach of all possible computers.
The classical Wallace–Bolyai–Gerwien theorem (1833) says any two polygons of equal area can be cut and reassembled into each other. That's decidable.
But when we move to collections of shapes, with constraints on how pieces may be reused and combined across shapes, the problem crosses into undecidable territory — via the same tiling mechanisms that Berger used to prove the undecidability of the domino problem.