The maths of 3D printing: what makes a set of $\mathbb R^3$ 3D-printable?

Question

Context (although it's not really important):

For a project, I may have to print some complicated 3D sets. The only thing I know about these sets is that they are defined as bounded parts of $\mathbb R^3$, are formed as a union of tiny balls, and may not be connected (I know that it is vague, but I'm also asking this question to modify the properties of this set to fit the criterion that allows a part of $\mathbb R^3$ to be 3D-printable).

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The question:

So I'm looking for references about maths modelisations of 3D printing (only classics FDM/FFF 3D printers). I want to see whether it is possible to approximate my complicated set with a 3D printable one.

So here I am looking for some reference about the maths of 3D printing in general. More specifically, I would like to answer the following questions:

1. What kind of vector space and topology may define a "3D printable" set? (or what kind of structure may define this properly?)
2. Given any bounded set, how to find the "best" (for some criteria that may depend on the problem) 3D printable approximation?

Further details:

1. For example, given a compact set over $\mathbb R^3$, what makes it 3D printable? To make more sense, what criteria it must meet to have a 3D printable set that is a good approximation? It seems self-evident that a non-connected set will not be 3D-printable. But are all connected bounded set printable for a high enough resolution? It also seems evident that it depends on the resolution and the precision of the printer. But how to characterise it mathematically?

$\longrightarrow$ According to the answer from @Ihf, they are generally triangulated polyhedra. I need some more information about it: what structure may we have over the set of triangulated polyhedra? Is there any good reference about it that I may read? If $\mathcal T\mathcal P$ denotes this set, is there any good description of $\overline{\mathcal T\mathcal P}$? (the closure of it in $\mathbb R^3$ as the usual metric space). This may give a good description of "3D printable objects" which might be $\overline{\mathcal T\mathcal P}$ (objects that are approximable with triangulation) or a subset of it (if we want some physical propriety to be verified).

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EDIT: I do not have any example, because of the reasons that I just added in the first paragraph of this post. I also edited the title to be more clear: I do not want to print something to illustrate some mathematical "classic" forms as suggested in a comment, but to print a set that is generated by iterated function systems for my case (I'm studying them in a general view, so I don't know yet which form I will try to print. That why I'm looking for criteria that it'd have to meet).

EDIT2: added further details for question 1.

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Final edit:

Since this subject seems not to be very documented with mathematical approaches, I accept the first answer I got, that answers my question.

So to sum up, the set $\mathcal{TP}$ defined in my post answer the problem. And thus, the well approximable sets are the ones in $\overline{\mathcal{TP}}$ since we can approach them with any precision with 3D printable sets.

I'm still interested though about good references on the mathematical description of 3D printing, so if you had any, do not hesitate to add another answer to complement the accepted one.

Answer 1

The 3D objects that are printable are typically polyhedral, usually triangulated. One of the standard file format is STL:

An STL file describes a raw, unstructured triangulated surface by the unit normal and vertices (ordered by the right-hand rule) of the triangles using a three-dimensional Cartesian coordinate system.

The whole field of geometric modeling is about your second question. The key point here is how a set is given.