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Circle becomes a pair of parallel lines

  1. Mar 19, 2012 #1
    Does anyone know if there is a non-Euclidean geometry(or something like that) where the circle becomes a line or a pair of parallel lines?


    ...or even where the circle becomes a set of parallel lines?

    ...i'm doing some work in Guitar Theory and this situation appears...


    Thanks

    The Roman Empire Guitar Research Institute
     
  2. jcsd
  3. Mar 19, 2012 #2

    HallsofIvy

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    It's hard to believe you are serious, but no, all different "geometries" respect the fundamental topological properties of sets. In particular, if a set is connected (as is a circle) in one geometry, it will be connected in any geometry (and "parallel lines" are not connected).
     
  4. Mar 19, 2012 #3
    I'm serious!

    But are you sure about your answer?

    I study a little Math also...I'm more on the Algebra 'side'...(i finally reached Group Theory proper...thank God!)

    BUT isn't there something about something becoming great circles....or something about great circles becoming something...etc...etc...?

    Isn't there a situation where a line becomes a great circle?


    Thanks
     
  5. Mar 19, 2012 #4
    If sounds like you're thinking of the geometry on a sphere--I've heard that geometry be referred to as "Riemannian" geometry, but I've also heard "Riemannian" refer to a much more general class of geometries. In the geometry on a sphere, the geodesics ("straight" lines) are great circles of the sphere. Doesn't that intuitively seem like the shortest way to get from one point to another?
     
  6. Mar 20, 2012 #5

    HallsofIvy

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    Yes, of course, in spherical geometry our "lines" are the great circles of the sphere. But that is still a connected set and has nothing to do with "the circle becomes a set of parallel lines" as in your first post.
     
  7. Mar 20, 2012 #6
    But isn't there a non-Euclidean geometry that says that 2 parallel lines may meet at 'infinity'?

    (Side: and if so, then isn't 2 parallel lines connected?------Connected at infinity???)

    Or am i just wishing for too too much?....i don't even have the definition for "connected".

    The situation: We have the guitar fretboard....we 'made' it into an infinite rectangular grid....we found the X and Y axes so every point/node on the grid can be located....the axes are not orthoganal....distances are measured in "semitones"....when we plot a circle it comes out to be a pair of parallel lines....when we compare other theorems to Euclid we realize that we have here some non-Euclidean situation going on.


    Any guitar mathematicians here?....ha.

    Thanks.



    The Roman Empire
     
    Last edited: Mar 20, 2012
  8. Mar 22, 2012 #7
    You're sort of correct, except keep in mind that its the same infinity in both directions, so we have four lines coming together at a point, and two lines spanning the plane. Kind of (exactly) like straps of a backpack wrapped around your body.

    What you're discussing is different, though! You're talking about metric spheres, (or pseudometric spheres, in this case) which are the set of points a given distance (number of semitones) away from a given point. It's not a "sphere" geometrically or topologically, it just has that property of containing all the points a given distance from the "center."

    You get two lines because each string is identical to the last, but shifted by a fifth (or fourth, depending on your direction). The "current" string has two points (count off your interval up and down. Each other string has those same two points shifted by five frets (except for that pesky g-b), so you get two points there.

    Your space is discrete (each point is "separate" from the others, rather than connected by a continuous array of points) so no subset of it at all is homeomorphic to a Euclidean circle. You do, however, have metric circles.

    On last point! The reason you have a pseudometric, not a metric, is becauase metrics require that "no two points lie at the same point." For any point, there are infinitely many other points (one on each string) that are "0 semitones away." In a metric space, only a point and itself may be 0 distance apart. This means that circles of radius 0 in metric spaces are just a point. In your space, the circle of radius 0 is a single line, since the two parallel lines come together.
     
  9. Mar 23, 2012 #8
    Yeah Alexfloo! You're correct on these points.

    This really helps me a lot.
    It really focuses my/the investigations a lot.
    It shows me some/the limitations of 'the situation'.

    I /we have been formulation this situation upon the guitar fretboard for decades now.
    We can write volumes.
    But you seem to have quite a hold on this situation!

    Some points:

    1) O.k. i see how to interpret 'infinity' now(...wonder if all mathematical realms interpret it this way?...)...i think remember seeing something like this interpretation in class...dealing with Reimann sphere stuff...
     
  10. Mar 23, 2012 #9
    Side: i just wrote my other points.....pressed 'submit reply'......then realized that i wasn't connected...

    so i lost my data!!!!

    will rewrite them tomorrow when i get some chemical energy back...
     
  11. Mar 25, 2012 #10
    So yes...

    2) Thanks. I did some research on the net and checked out definitions of 'connected', metric space...etc.....And indeed, it is a metric space OR a pseudometric space that i/we have stumbled upon!!!!!!!!

    my problem is/was that i've been calling 'the object' a line(continuous) now for so so long that i actually came to believe that it really was a line. Ha. Fancy that.

    Thanks for awakening me...ha.

    But there's no secret formulary for transforming the discreet into the continuous YET!!!!!!!...ha.



    3) If we interpret the situation as being collapsed onto only 1 string.......we indeed have a metric space......maybe this is an obvious 'trivial case' and i'm making a fool of myself..?

    If we interpret it on the usual 6 strings(or on the infinite plane of strings(maintaining the original ordering on the guitar fretboard..etc..)) we have a pseudometric space. Indeed.

    Thanks HallsofIvy & Alexfloo......sorry for thanking you guys so much but this realization za'gonna save me a lot of twirling around in circles.


    4) Here is the distance formula for the metric/pseudometric:

    If P1 is (x1, y1) and P2 is (x2, y2) ----> d = abs(x1) + abs(x2) + abs(y1) + abs(y2)

    abs(x).....absolute value of integer x.

    If anyone has any information on such a distance formula we'd be quite happy to have it....ha.


    ...etc

    Thanks


    The Roman Empire Guitar Research Institute of Montreal, Canada
     
    Last edited: Mar 25, 2012
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