Where to start for studying geometry and what way should I follow after the first step?

I'm reading Gruenbaum's Tilings and Patterns:

It's curious that almost all aspects of geometry relevant to the "man in the street" are ignored by our educational systems. Geometry has been almost squeezed out of school and university syllabuses and what little remains is rarely of any use to people who wish to apply geometric ideas in their work[...]

I kinda felt that when I was in high school, and once I've seen the proof which states that an exterior angle of a triangle is greater than either remote interior angle in euclidean geometry and I felt it was kind magic, it was another way of thinking, I don't know how to explain but this experience made me get interested in geometry.

So I'm curious on what I should start studying/reading for learning geometry and what way I should follow, I know that euclidean geometry is one of the first geometries arround, but I've read somewhere that there's a discrepancy between euclidean geometry and the modern way of looking at geometry, so I'm not sure on where to start.

My question is a little extense, I'm trying to figure out where to start studying geometry and I'm trying to find a reasonable way to follow and also books and video lectures for it, I want to use this question as a reference for the next years and it also would be useful to MSE members to find a guide into the study of geometry. Can you help me?

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You can try Stillwell's The Four Pillars of Geometry. It doesn't assume much background. I have used the book before to read about hyperbolic geometry and it was very helpful. – user38268 Mar 24 '13 at 6:24

1 Answer

There's quite a bit of geometry in typical mathematics courses, if you know how to look for it. The problem (in my opinion) is that a lot of modern mathematics has lost its connections to geometry. Why? Because drawing good pictures is difficult, because "pictures are not proofs", and are therefore discouraged, because things are done in highly abstract spaces rather than plain old 2D or 3D, because abstraction is "elegant"? Who knows.

Anyway, take courses in differential geometry and algebraic geometry. Try to understand the geometry of linear algebra and complex variables. Whenever you see an equation, ask yourself what kind of geometric object it represents. If you're studying things in spaces with $n$ dimensions, try the cases $n=2$ or $n=3$ first. Draw pictures, even if your teachers discourage you.

This book by Needham is an example of how much geometry you can find if you look for it.

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