I wondered if back in the time of ancient Greeks mathematicians, 3D geometry was discovered as result of plane geometry? (Was there anything in the axioms of plane geometry that indicated existence of 3D geometry?) or was 3D geometry a familiar concept already and there were simply attempts to find its axioms?
Basically, at the time of ancient Greeks mathematicians was geometry discovered mathematically or was it familiar concepts for which axioms were derived?
Three-dimensional Geometry is what the last three books of Euclid's Elements are about. No, it was not discovered mathematically, at least not in sense that it was seen as a generalization of two-dimensional Geometry. Already in ancient Egypt and Babylon it was known how to compute the volume of a truncated quadrangular pyramed, for instance, long before axioms were introduced in Geometry. What Euclid added was an axiomatic approach to an already known subject, besides, possibly, adding a few new theorems of his own.
It is not known how two-dimensional or three-dimensional geometry were discovered. All we have are some legends recorded by the people who lived centuries after these discoveries. It is quite plausible that 2-dimensional and 3-dimensional geometry were discovered simultaneously.
One of the legends says that Thales measured the height of an Egyptian pyramid. It gives no detail about how exactly he did it, but it is quite possible that he had to use some geometry in space. (Thales was probably the earliest mathematician on whom we have any record, but his work did not survive and details which are mentioned by later writers are unclear).
It wasn't really discovered in a mathematical sense of "exploration of higher dimensions" but more as a result of everyday experiences.
Euclidean "plane" is classic 2D Geometry - studying what can be done with lines/triangles and circles, the primitive geometric elements, and deriving various proofs. This is not clear from everyday experiences.
You don't find perfect triangles in nature. They're a geometrical construct that uses the concept of a line (which itself is not "well defined" by modern standards in The Elements). So, the study of 2D geometry does require creative thought and application of deductive logic/proofs, constructions with straight-edge and compass (for the most part).
However, we see the world in 3 dimensions. To give "perfect meaning" to shapes, they used the 2D analogs and built what we call "surfaces of revolution" to define some basic shapes they were familiar with (mostly owing to their experiences with pottery perhaps):
In theory you could just rotate any primitive shape and get a corresponding 3D shape. We've lost Euclid's book on Conics so we don't know if they ever rotated those to get some interesting shapes. The cube was just the most primitive shape. But I'm sure they made some vases in those days so I'd conjecture they did something :)
They studied some "other shapes" like the platonic solids, the various "hedrons" (tetra-, dodeca-, icosa- ...) since they combined symmetrical shapes to get interesting 3D objects.
So, they just analyzed the shapes and explored the corresponding 2D counterparts i.e., areas in 2D implied volumes in 3D. They concluded that volumes are proportional to the "cube" of the "variable" i.e., length of radius/side of a cone/cube, since areas are proportional to the "squares" in the 2D world. Finding exact formulae was a separate thing (e.g., for a sphere)
Eudoxus had conjectured that the volume of a pyramid is $\frac{1}{3}$rd that of the enclosed "cylinder" i.e., a cube. Euclid 'perfected' this method of exhaustion as we call it, and published a clean proof in his books.
If you read the elements you'll see, that 3D analysis was a natural extension owing to everyday experiences. They proved what they could and what made most sense "by analyzing" shapes or cutting them (e.g., you'd always get an ellipse if you cut a cylinder diagonally across it's height).
The advantage of this "study" was applications to architecture I believe but nothing beyond that. The greeks firmly believed in a world they could "perceive" and hence they spent time analyzing that, even in their mathematical endeavors. But I don't think they did it as an exploration of higher dimension math.
