I’ve recently become somewhat interested in smooth manifolds and surfaces as a result of preparation for various interviews, amongst other reasons. The concept of a surface is very intuitive, and is a concept that a student of mathematics is likely to encounter very early in their mathematical careers, though a reasonable definition of a surface takes more effort. The result of this formalisation is a remarkably elegant theory, which eventually leads to ideas of smooth manifolds – arguably one of the best sounding mathematical objects – and generalised calculus.
It took quite some time before I had a “big picture” of what a surface is, and how this fits into the grander theory. Undoubtedly I am missing some of the major pieces to this puzzle, but it does show some of the elegance of the theory (at least for me).