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Excitment of geometry?

  1. Nov 15, 2007 #1
    I am not very thrilled by geometric/topological concepts. Is anyone is? If so why?
    If anyone is like me then provide reasons as well.

    Anyone know a book or two on geometry/topology that might bring excitment into someone who didn't see geometry as intersting?
     
  2. jcsd
  3. Nov 15, 2007 #2
    Admitedly, I never really thought that classical geometry was very interesting until I started learning about differential geometry and galois theory. I had originally held disdain for what felt like trivial or flimsy arguments concerning the length of a side, or the angle between two vectors and so on and so forth. However, upon learning of the aforementioned topics, and the ability to qualify geometry in a way that made it much more fun and interesting to play with was what changed it for me.

    Some mind blowing things were how we can use Galois theory with regards to things like compass and ruler constructions, or how given a parameterization of a surface we could find its curvature; how the sums of angles of a triangle in non-Euclidean space. Things like these are what's changed my mind.

    As far as topology, I was traumatized by having a poor teacher during a burn-out term, and so I fear it will take quite a bit to get me back to it.
     
  4. Nov 15, 2007 #3
    I haven't done differential geometry. The compass and ruler constructions dosen't seem to be that exciting either. I prefer galois theory as a algebra theory without reference to visual information.

    My dislike for geometry can be summed up in one word: messy. I think geometry is quite messy. It could be that I am missing some key ingredients though.
     
  5. Nov 15, 2007 #4

    robphy

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    Geometry makes General Relativity easier to understand.
     
  6. Nov 15, 2007 #5

    mathwonk

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    have you ever seen euclids construction of a regular pentagon?
     
  7. Nov 15, 2007 #6
    "The Shape of Space" by Jeffrey Weeks is a great book on topology and geometry. If it can't make you excited about those subjects, nothing can. It also has a great problem set integrated into the text and a complete set of answers (do the problems before looking at the solutions!). Weeks has a Phd from Princeton and really knows his subject.
     
  8. Nov 15, 2007 #7

    JasonRox

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    Do we want to? :smile:
     
  9. Nov 17, 2007 #8
    Yes! If you care to see beautiful mathematics.

    Another good book is "Journey Through Genius". It has some wonderful geometric proofs.
     
  10. Nov 17, 2007 #9
    I've read that book. Some nice geometries in it yet but I still dislike it. Maybe I have to do more excercises to get used to them. Riemann infact disliked geometry as well at first (I wonder why, probably haven't done enough of it at the time?) but Gauss pushed him to do it and Riemann revolutionsed the subject.
     
  11. Nov 17, 2007 #10

    Hurkyl

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    The elementary theory of Euclidean geometry is equivalent to the elementary theory of real numbers...
     
  12. Nov 17, 2007 #11

    mathwonk

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    the classical theory of euclidean geometry is much deeper and more difficult to do properly, i.e.with full proofs, than is usually thought. extremely few people ahve ever seen a thorough development of euclidean geometry, which i find quite beautiful and stimulating.

    the best soiurce is euclid himself, but it helps to have a guide, like the marvellous book of hartshorne: geometry, euclid and beyond.

    the theory is actually far more complkex than that of the real numbers, as there s no need to assume the dedekind axiom. thus euclidean geometry works with subfields of the reals, closed under certain operations corresponding making various geometrical constructions.

    hilberts axiomatic foundations are perhaps the best, where he explicitlky states he will not use dedekinds axiom in order to be able to make conclusions about numbers from th geometry, instead of vice versa.

    the system of birkhoff does assume each line is equivalent to the reals, but birkhoff is not doing strictly euclidean geometry, since he is assuming also a unit. thus the group of automorphisms in birkhoff's system excludes scaling.

    i have been very challenged and enlightened to present merely a basic course in neutral and euclidean geometry this semester, with due attention to all details of proof. in particular it is interesting to keep straight which construtions and theorems are true withiout the parallel postulate, and hence hold in hyperbolic geometry.


    for instance it is known the concurrence theorem for medians of triangles is true withut the parallel postulate, but the noly proof i have been shown, uses the riemannian manifold structure of the geometry. i challenge you to find an elementary proof in ordinary elementary geometry language.


    euclidean and hyperbolic geometry are the first examples of riemannian manifolds, and it is of great interest and not at all trivial to consider the differentiability of the dependences in geometry. e.g. the fact that SAS implies congruence means that the length of the third side of a triangle is a function of the lengths of the first two, and the angle. one can ask if this dependence is differentiable, and in particular how to compute the partial deriavtives of this dependence.

    this idea is the basis for the proof i have been shown by robert foote of wabash college.

    i do not know how to answer the question posed in this thread of interest in geometry. usually the problem of low interest in a subject stems from a bad introduction in a bad course. the material itself is always fascinating if exhibited by someone who knows and loves it.

    i used to hate real analysis, but when i hear an expert analyst, and an expert instructor, speak about it i am usually entranced and instructed.

    i also used to dislike the algebraic formulation of geometric ideas in abstract algebraic geometry, but after years of patient instruction from robert varley, and reading books by mumford and hartshorne, i find it beautiful and deep.

    a first instance of this interaction is the field theory formualtion of constructibility, which builds on the insight of descartes, as hartshorne makes clear, and uses, not galois theory, but merely the dimension theory of field extensions, as one of the early directors of MSRI originally pointed out to me. (A senior moment hides his name, a famous ring theorist.)...Irving Kaplansky.

    some other comments have been trashed by the fickle browser.
     
    Last edited: Nov 17, 2007
  13. Nov 18, 2007 #12
    I personally fell in love with algebraic geometry after I read from Miles Reid's text.
     
  14. Nov 18, 2007 #13

    mathwonk

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    Last edited: Nov 18, 2007
  15. Nov 18, 2007 #14
    His commutative algebra book?
     
  16. Nov 18, 2007 #15
    Yes. It's not really commutative algebra. It's commutative algebra viewed wholly from an algebraic geometry perspective. The heart of the book is looking at the affine varieties in algebraically closed fields.

    You need to know a fair bit of ring theory before I could recommend this book to you.

    Thanks for the linky wonk.
     
  17. Nov 18, 2007 #16

    mathwonk

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    reid also hasan undregrad alg geom book, thats what i thought you meant. you should begin there.
     
  18. Jan 17, 2008 #17
    I am looking for a more general introduction to geometry. Has anyone read 'Four Pillars of geometry'? The good thing about that book is that it introduces geometry not as one single thing which one must grasp but different things. i.e I was surprised that the mathematics via the cartesian plane (a standard in high school) is considered as geometry. I use to think that Euclidean (in the strict sense of the ancient Euclid) and noneuclidean geometry were the only geometries. I think that this book may have brought some excitment in me. Anyone else tried it?
     
  19. Jan 17, 2008 #18
    Yes but topology only makes it harder!
     
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