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Is it worth studying geometry during my summer?

  1. Feb 9, 2015 #1
    I am a first year computer engineering student. Having programmed several years, I realize my math skills are not adequate for my liking (planning on improving them before tackling some more advanced data structures and algorithm books).

    My plan is to read how to prove it, spivak calculus, a probability book, and maybe something about linear algebra in the 4 months I have off. (I already have taken linear algebra, but I'd want a more theoretical book)

    I've been considering replacing one of these books with learning Euclids Elements, as I noted that Newton's intellect peaked after reading this book. As someone who has never taken geometry before, would it be worth doing geometry in my summer for my major?

    I plan on studying very hard, 10+ hours a day, 6-7 days a week.
     
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  3. Feb 9, 2015 #2

    Bystander

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    Geometry is always worth studying.
     
  4. Feb 9, 2015 #3

    SteamKing

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    I would recommend that you start slow and study the basics before jumping into a text like Spivak. For studying calculus, even the basic stuff, you should be up on your algebra, geometry, and trigonometry first. If you haven't studied geometry before (what math did you do in HS?), you probably don't know any trig.

    Studying 10+ hours a day, 6-7 days a week is an admirable goal, but I think you are going to find it's a bit of a grind after a few days. Fatigue is not conducive to learning new concepts, especially if these concepts have any complexity of subtlety at all. I know there's plenty of time to sleep after you're dead, but you should budget your time for some trivial things, like eating and sleeping, if not bathing regularly. You need to give you mind a bit of a rest from studying too much.
     
  5. Feb 10, 2015 #4

    micromass

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    Euclid's elements is a very outdated book. Sure, it is a classic and many great minds studied the book. But there are many better books out there, both in content as in pedagogy. I personally find Euclid quite difficult to read, and modern treatments can be way more illuminating.

    Also, geometry (the way Euclid did it), will not be very useful for you. So if you're interested in geometry for its own sake, then by all means: do it. If you're hoping to get some useful skill out of it, then you should probably not do it. The geometry you really should master is trigonometry, linear algebra and very basic Euclidean geometry.

    I hope I don't come off as a Euclid-hater here, since I do recognize his enormous influence on mathematics.
     
  6. Feb 10, 2015 #5

    QuantumCurt

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    I own Euclid's Elements for the cool factor, but as an actual tool with which to learn geometry today, it pales a great deal in comparison. It's not a textbook as such. There are quite a few other choices that would be far better.

    It's admirable to want to learn so much, and I can definitely relate. However, you've named some difficult stuff. Spivak alone is a large undertaking. The odds of actually being able to work through all of those books alone in four months and actually retain a meaningful amount of information isn't great. One would be better off working through one or two of them. If you're worried about your geometry skills being up to par, then you're probably not ready for Spivak.
     
  7. Feb 10, 2015 #6
    I did all the math/science courses offered by my high school (calculus, physics, chemistry, biology). I'm currently taking calculus 2 right now in my second semester. I've also completed linear algebra. I did study trig, and I use basic geometry all the time (such as angle bunny hopping, and I guess cutting things up into triangles). I also do eat, sleep, and bathe regularly. I normally spend my summers coding/doing security for long hours. But I'm going to jump into more math this summer. This prior summer, I must have spent on average 4-5 hours a day doing math/physics/generally reviewing everything to prepare myself for calculus 1. However, the math I'm taking is engineering math so it is not very proof intensive.

    I'm not exactly sure if my geometry skills are up to par. For instance, if doing courses like calculus 2, linear algebra, dynamics, mechanics, electricity, etc would qualify me as having good geometry skills, then I think I might be up to par.
     
  8. Feb 11, 2015 #7
    I think there are some theorems from geometry that are useful to know to gain more intuition about certain things, like some of the facts from complex analysis. And in general, it's a good way to develop some visual intuition. So, I think it's worthwhile. I'm not sure Euclid is the best place to learn it from, though. I like Lines and Curves: A practical geometry handbook.
     
  9. Feb 11, 2015 #8
    You should definitely learn geometry! It trains your brain to think in new ways, and is a fascinating subject. Some theories of physics, like the "General Theory of Relativity," are impossible to understand without geometry. You won't regret it!
     
  10. Feb 11, 2015 #9

    symbolipoint

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    Studying Geometry during the summer is worth the doing if you can study this 10 to 12 hours per week EVERYDAY. You will most likely not finish, but the study can be very much worth doing.
     
  11. Feb 11, 2015 #10

    mathwonk

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    I also found euclid difficult to enter for many years, but admittedly only began it briefly, and was turned off by the definitions, which i failed to understand, and left it aside for decades afterwards., But in my 60's, with the guidance of a great book by Hartshorne, I tried again to read it and found it by far the best geometry book in existence. I am a professional algebraic geometer, and in my recent experience with euclid i discovered that he had given essentially newton's definition of a tangent line as a limit, had given essentially dedekind's definition of real numbers as least upper bounds of rationals, and m,any other beautiful and deep ideas.

    I do recommend however using a guide, such as Hartshorne's Geometry: Euclid and beyond.

    https://www.amazon.com/Geometry-Euc...TF8&qid=1423717670&sr=1-3&keywords=hartshorne

    A free introduction to the topic can be found in this essay by him:
    http://www.ams.org/notices/200004/fea-hartshorne.pdf

    my own free notes on euclid, written in connection with a 2 week course i gave to brilliant 8-10 year olds is on my website:
    http://www.math.uga.edu/~roy/camp2011/10.pdf

    in my opinion, euclid is the best math book you can possibly read. it lays the foundation for all math to follow for centuries.
     
    Last edited by a moderator: May 7, 2017
  12. Feb 11, 2015 #11
    I agree that the contents of Euclid are good, but I think it tends to be a little too formal to be the optimal thing to learn from, which is why having a guide like Harteshorne would probably be a good idea, although I did okay just having the online version with commentary here:

    http://aleph0.clarku.edu/~djoyce/java/elements/elements.html

    I never made it all the way through the Elements due to lack of time, although I did skip ahead to the last chapter and understood a lot of it. It's pretty interesting because it covers the Platonic solids.

    I focused a lot on all the ruler and compass constructions when I was reading it, and developed a lot of my own visualizations to explain the proofs. A lot of times, I used the old approach of trying to prove the theorems for myself before reading the proof. I found it fairly understandable, but I could learn it faster from a more friendly source. And I'm pretty good at coming up with my own intuition for things.
     
  13. Feb 12, 2015 #12

    mathwonk

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    here is the beautiful copy of euclid i used and recommend:

    https://www.amazon.com/Euclids-Elem...=1423762615&sr=1-1&keywords=euclid's+elements


    In particular I found the lengthy one with commentaries by Heath rather tedious. I prefer sticking to the master's words rather than the commentaries on them. The book by Hartshorne worked for me first of all because he declined to paraphrase euclid, but simply assigned readings from it, and that forced me to actually open euclid and look inside after 40 years.
     
  14. Feb 12, 2015 #13

    lavinia

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    I think Euclidean geometry is indispensable for understanding much mathematics. It is an attempt to define the primitive notions of space and of geometric ideas.
    Also it has played a huge role in mathematical history. Much of the development of modern mathematics is linked the the discovery of non-Euclidean geometries and of new ideas of space and geometry.

    Euclidean geometry introduces the idea of proofs which is also indispensable for mathematics.
    I have never read Euclid's elements but however you learn this stuff, it is probably helpful to think about the underlying ideas of what space is like and how one defines geometric ideas.
     
  15. Feb 12, 2015 #14

    mathwonk

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    the treatment of area and volume in euclid is especially interesting. he treats equality of plane areas by finite equidecomposition, but for volumes he introduces a use of limits. The idea is that if two figures can be approximated arbitrarily well by figures of equal volume then they also have the same volume. a related method is used to treat similarity of triangles, wherein it is argued that the ratios of two pairs of segments are equal if every rational number less than one is also less than the other.

    Although limits are introduced, calculus is not. I.e. the fundamental principle used to calculate volumes in calculus is that two figures whose slice areas are equal at every height, also have the same volume, is a more systematic way of approximating the volume of figures than used by euclid. this method, called now cavalieri's principle, is due to archimedes, and can be found in his writings. nonetheless euclid starts the process of finding volumes by limits.
     
  16. Feb 12, 2015 #15
    I never needed any forcing. I just read it, was fine with it, but thought maybe I'd absorb it faster if the ideas were explained more clearly. For example, Euclid tends to label all the points and talks about line AB, rather than labeling the line by a letter like a, as a modern book might. So, it can be more cumbersome than it needs to be at times.
     
  17. Feb 13, 2015 #16

    QuantumCurt

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    At what level is Euclid's Elements appropriate? I have a copy of it, but I've never spent any considerable time with it. I'm considering taking a 400 level Euclidean Geometry course in my junior or senior year. Would this be comparable to the level of treatment at that level?

    The copy that I have is an older one that doesn't seem to be translated well. I've considered getting a newer copy. I like the looks of that Heath translation.
     
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