How to prove that these three distinct "recursive geometrical constructions" converge to the same object?

Question

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 prove 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?
Prove 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:

Here Obj1 in fractal form to compare with Obj3:

Without precise mathematical statements of undefined terms, and precise mathematical statements to be proved, there's not much of anything that can be said here. For instance, your concept of "isomorphism" is undefined. Also, your objects are not defined: the only thing in your post about those objects is some pictures and some external links. Also, your questions are overly vague; asking for us to formalize your musings is off-topic. — Lee Mosher, Aug 28 '25 at 13:05
I suggest, in accord with our suggestions to ask one question per post to give definitions, that you edit your post to be about one favorite "object", with a full detailed definition. I also suggest that you ask a practical, carefully stated mathematical question. — Lee Mosher, Aug 28 '25 at 13:05
And one thing to keep in mind: where you write "a mathematical proof is necessary", I would add that a mathematical statement is necessary in order to know what to prove. — Lee Mosher, Aug 28 '25 at 13:08
Hi @LeeMosher, thanks. Yes, I see now, it's complex, so we can reduce the question to "obj1 vs obj2". If you agree I can edit the question to simplify it. About "better definitions", there are many alternative formalisms to describe and compare objects... I'm citing references via links. For more details, see my Wiki Answer, we can edit there, and I can supply more subsidies for your answer. — Peter Krauss, Aug 29 '25 at 00:52
For @LeeMosher, and readers who prefer a more objective question, here is a reformulation of the domino problem https://math.stackexchange.com/q/5093964/70274 — Peter Krauss, Sep 03 '25 at 10:38