It seems like a topological isomorphism problem ... I'm starting with an "intuitive isomorphism recognition" of 3 objects, which are defined by geometric construction. I'm looking for a formalization sketch, if there is no obvious solution to proof the isomorphism.
The objects to be compared are aperiodic tilings of dominoes, and they exhibit the behavior of domino tiling patterns. Each object is using a different construction technique:
Obj1 is knowed as was "table tiling" builded by geometric cut-and-project. It is also in described this paper , as a classical domino tiling.
Obj2 is a obtained from a Hilbert curve transformation, described here by merging adjacent cells with the rule $j = \lceil{i/2} \rceil$.
Obj3 can be named here as "Munkres's fractal" because we can found at "Theorem 44.1" into this Munkres's notes.
In the initial construction process of each object, they are different and is difficult to check similarities. They are all recursive processes and quickly "converge" to the same pattern, that I am supposing that is also an isomorphism. The question can be splitted into two "how to" parts.
How to:
Formalize all the construction processes in the same framework?
That is, what the better theory/approach to describe all them with the same language?Proof that all converge to the same, where I am supposing that they are isomorphic.
... For example expressing all objects by L-system models, them showing that, by adjusting parameters and rule names, they are equivalent models.
The objects
Objcts 1 and 2:
and Obj3:
