The 'in general' part is important: of course, you can trisect specific angles like a 90 degree angle.

(The construction of number field extensions is happening because let's say you construct a 90 degree angle, well that allows you to construct an isosceles triangle with equal sides length 1, and therefore you construct the square root of 2, so the extension Q(sqrt(2))

Those are the special points you are interested in when you want to distinguish between vertices and other points on the boundary.

But I would say as a mathematical process, pretty likely once you have the basic ingredients. Richard Dawkins has written some good popular science books like 'Climbing Mount Improbable' that talks about evolution in this light. His 'Selfish Gene' is also an excellent read. It is slightly more technical but definitely understandable without an immense technical background.

As you begin to do so, you can keep a record of them in a spreadsheet or notebook. Look up their biological classifications. All of a sudden the classification system will start to make sense, and help you understand the diversity of the organisms around you.

Biology is one of those subjects where part of the context comes from being outdoors and trying to understand ecological relationships yourself. You don't have to get to the level of a pro biologist, but there is so much you can find out by experiencing nature and trying to understand the basics that anyone can do it.

It's much easier to understand if we take an example. An example of a theory is the single sentence:

"There exists an X and there exists a Y such that X is not equal to Y."

(Of course typically in logic you would use logic symbols, but here I am writing out in an English sentence.)

Now, a model of this theory is the set {1,2}. Another model is the set {1,2,3}. More generally: any set with at least two elements is a model of that theory. The "function symbols" and "relation symbols" can be introduced in the language to talk about operations like addition and multiplication.

For example, the theory of groups uses the language of groups with a binary function symbol representing group multiplication. Any group (such as the integers with addition or invertible matrices with matrix multiplication) is a model of that theory.

So: theories are sets of axioms in some language, and models are sets together with actual functions/relations that satisfy those axioms.

Models of ZFC are a little bit counterintuitive. But they are single sets that interpret all the axioms of ZFC, rather than actual sets that we use in informal mathematics. Models of ZFC can be quite unusual because of the incompleteness theorem, and there are infinitely many models because of this (such as some in which CH is true, etc.).