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IF i was a 1 dimensional being living in a 1 dimensional Universe being a circle.

  1. Jan 19, 2012 #1
    How could i mathematically proove that im living on a circle. Almost got it last night , just need an insight to figure out an equation.

    Ty.
     
  2. jcsd
  3. Jan 19, 2012 #2
    Circle is object in two dimensions!
     
  4. Jan 19, 2012 #3
    the circle being 1,2,3,4,5 or 6 dimensions is not important. I need the maths to get me there.
     
  5. Jan 19, 2012 #4

    marcus

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    I like the question! In differential geometry you do not need an n-dimensional space to be embedded in a space of higher dimension. Probably I disagree with Marina. You can define what the 2-sphere is without it having to live in a 3D environment.

    So a ring can be defined and studied without assuming the existence of any surrounding 2D space.

    However it is trivial. there is no intrinsic way to define a curvature that could be non-zero. There is (at least in my opinion) no way that the 1-dim creature can discover the topology of his world unless he makes a voyage of exploration. He must circumnavigate his world.

    He must tell his 1D friends to stay at one place, to mark that place, and then he must travel as far as necessary in one direction. when he sees his friends again he will know that it is a ring.
     
  6. Jan 23, 2012 #5
    surely a ring has more than on dimension, two at the least

    A 1D universe would just be a line if a 1D being did try to determine the shape of its universe by marking a position and seeing if he came back upon himself suggests that the universe is 2D or greater. A 1D universe would not have a shape to it. Or am I mistaken? show me a shape that has 1 dimension

    As for the Op, I have no idea how a 1D being would be able to prove he is on a 1D universe using mathematics

    EDIT - It has come to my attention that even a line is a 2D object. I cannot see or imagine how a 1D universe would work, it would have no shape in any sense of the word, making your question impractical.
     
    Last edited: Jan 23, 2012
  7. Jan 24, 2012 #6
    You can have a one-dimensional line that is effectively looping back on itself by just identifying two endpoints of the line segment and thus giving it non-trivial topology. This does not require another dimension.
     
  8. Jan 24, 2012 #7
    Does anyone have any links that can support this sentence?
    I do not see how a 1 dimensional universe can loop back on itself without the presence of an extra dimension

    I imagine a 1D universe to consist of just length, without width or depth, without these dimensions their would be no discernible shape to it.
    Have I misunderstood something here?
     
  9. Jan 24, 2012 #8
    Yes, you are missing out topology and a whole field of mathematics known as differential geometry. Modern differential geometry basically started with Riemann's realization in the 19th century that we can talk about geometry *intrinsic* to a surface without embedding it in a higher dimensional space. Perhaps this will help.
     
  10. Jan 24, 2012 #9
    But how does this apply to the Op's question and a 1 dimensional existence.
    How can there be topology in the 1D
     
  11. Jan 24, 2012 #10
  12. Jan 24, 2012 #11

    phinds

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    Thanks for that. Helped me see the possibilities in a way I wasn't thinking of them. I was stuck on the concept that the 1D universe had to be an open line segment, somehow didn't wedge it into my brain that a circle is a valid version of a 1D universe.
     
  13. Jan 24, 2012 #12

    marcus

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    For a math major in the 1960s one's first exposure to topology was the topology of the line (the real numbers) which of course is 1 dimensional. Many of the definitions can be illustrated. Topology is a rich field.

    Thanks yenchin! also for your reply in this other thread, where Chalnoth also joined in with some helpful amplification.
    https://www.physicsforums.com/showthread.php?p=3724707#post3724707

    As I see it, some of these elementary math questions might be better pursued in the math forum. Or in Relativity forum. After all, cosmology (the topic here) is based on GR which in turn is based on differential geometry as developed in the 19th century by Riemann.
    If you want to discuss the basics of the math foundations---the language used to formulate cosmic models---differential geometry---maybe this is not the place.
     
  14. Jan 24, 2012 #13
    Thanks for clearing that up for me.
     
  15. Jan 24, 2012 #14

    marcus

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  16. Jan 24, 2012 #15
    Remember, a line does have only length, even if it happens to be a line that loops upon itself (think the circumference of a circle), a 2 dimensional object has area (think the surface of a sphere). A 3 dimensional object has volume, etc. The only reason some might think that a circle (or any line) is 2 dimensions is that we are viewing it from the perspective of 3 dimensions.
     
  17. Jan 27, 2012 #16

    alt

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    How can a straight (one dimensional) line loop around anything ? Doesn't make sense to me. Or has 'straight' been redefined to mean 'slightly curved' ?
     
  18. Jan 27, 2012 #17

    phinds

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    Why are you imposing "straight" on the concept of one-dimensional? The OP correctly did not.
     
  19. Jan 27, 2012 #18
    Don't even think of a shape at all.
    Takes the numbers 1 to 10. That is a straight, 1 dimensional line.
    Now when you count to 10, what's next after 10? Stop or loop back to 1?

    I'm not sure what a 'not straight' 1d line would be like...
    Just a guess, but I would suppose it would be a non-linear line. As in not all points are evenly distributed. Count from 1 to 10 but skip 3,4 and 7.
     
  20. Jan 27, 2012 #19
    Think of a straight line in cartesion coordinates, say, y = 2. The equation of this is a constant function, with no value that varies (for every x value, y is the same). Now, take a circle in polar coordinates, defined by the equation r = 2. This is a constant function, also with no value that varies (for every theta, r is the same).

    I think the problem is that there are two distinct ideas of dimension that are being used here. One is mathematical, and the other is visual. People probably tend to think in terms of cartesian coordinates, so anything they think of is visually embedded in the 3-d certsian system, and a circle "visually" to us is in 2 dimensions (say, x and y). However, in mathematics,, and specifically differential geometry, as has been pointed out, it is unnecessary to embed any curve in a higher dimensional space, so that a "line" in some arbitrary space is defined by the curvature of that space.
     
  21. Jan 27, 2012 #20
    I was of the opinion that we were talking of visual dimensions, but that's been cleared up for e since, i was unaware of topology and all that jazz, but now i am I have plenty of studying to do in my spare time now. Thanks to this thread. If anyone can provide more reading material on this subject i would really appreciate it.
     
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