Exploring Geometry: A Modern Approach for Self-Learning

[generic_]

Well I have been teaching my self mathematics for about 4 months now. Started with pre-algebra now I'm on "modern algebra". Thing is I haven't taken geometry in high-school(and will not be able to). So I'm looking for a quick book that is interesting enough to keep me well, interested. I know basic geometry of-course and any geometry problem I've seen so far has not been a problem for me. But now that I'm starting to get into higher math some books are showing "geometric proofs" and I have looked farther into the book some of these get too much for me. I don't really like geometry much, every time I try to teach myself I end up doing algebra or working on sets. Even though I consider myself more determined than most , I'm not at a really high level of mathematical maturity so books like Euclid's Elements are too much. Preferably I'm just looking for an interesting book, it does not have to be a quick book, just something that doesn't put me to sleep. Any suggestions?

 

[quasar987]

Extremely intesresting is the book "Geometry and Topology" by Reid and Szendroi. It is about geometry in the modern sense however meaning that the authors are working with coordinates (Descartes) rather than with axioms (Euclid), and that plane (aka Euclidean) geometry is treated on an equal footing as the other geometries: spherical, hyperbolic, affine and projective.

 

[generic_]

That is most definitely a very interesting book. It covers a wide variety of topic in which geometry is used, But most of the book is well above my head, though from what I saw of the first chapter it is good. This won't be of much help to be honest I haven't even covered basic geometry yet. It has gotten me a little interested though : ) Though this book will definitely go on my list.

 

[sganesh88]

This one is good.Goes from very basics.

 

[Emustrangler]

There are two basic approaches, the modern "analytic" one that starts with the real number plane and derives results (as in the book quaser mentions), and the more classical approach using geometric axioms that Euclid and most high school classes use. Personally I think the latter is more fun and probably also easier to find basic books on.

Euclid's elements Books I, III and IV and V might be all you need, and aren't really that high level (indeed, they're not really rigerous by modern standards, even though they're usually held up as an example of mathematical rigour.)

Plus you can find them online with notes here: http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

I remember liking Hartshorne's Geometry: Euclid and Beyond which develops Euclids Elements from Hilberts axioms (so its more rigorous, but still classical in nature and accessible to HS students). The latter chapters need some abstract Algebra, but you should be able to read most of it without any higher maths, and its written at a friendly level.

Good luck.

 

[MathematicalPhysicist]

Another good suggestion is reading any of Coxeter textbooks on Geometry. (if I remeber it's 3 volume books series).
He's most notabely known as the last classical Geometer, i.e modern geometry is mainly the synthesis of algebra and classical geometry.