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B 1D space in 2D space

  1. Jul 25, 2016 #1
    Wikipedia says this:

    "the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it."

    Say that there is 1D space "contained" within 2D space and the former can be represented as a line in a 2D Cartesian coordinates system. I am under the impression that the line that represents 1D space must be a straight line or else it will "extend" into 2D space and need more than one coordinate to specific any point in it. At this stage (according to the defination above) the line would not be one dimensional.

    Am I correct?
     
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  3. Jul 25, 2016 #2

    phinds

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    If you draw a sine wave on a piece of infinitely long graph paper, does it extend out of the paper?
     
  4. Jul 25, 2016 #3
    No it wouldn't but then you'd need more than one coordinate to define a point on that graph which would mean that the line wouldn't be one dimensional.
     
  5. Jul 25, 2016 #4

    phinds

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    No, the line is still one dimensional. If you are in the world that consists of only the line, you have a single direction in which you can go forward or backward. The IS no "up/ down" or any other direction so it IS one dimensional.

    EDIT: by the way, this is a very common misconception that you have.
     
  6. Jul 25, 2016 #5

    mfb

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    You can use a single coordinate, e. g. distance from the origin along the line.
     
  7. Jul 25, 2016 #6
    But what happens if a 2D observer sees the line from 2D space?
     
  8. Jul 25, 2016 #7

    phinds

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    That is irrelevant to whether or not the line is one dimensional. Remember, you HAVE to think of existence in the world of the line with nothing else existing. There IS no "2D" to the constraints of the world of the line.
     
  9. Jul 25, 2016 #8
    But surely if the line is "contained" within the 2D Cartesian coordinates then there is 2D constraints.
     
  10. Jul 25, 2016 #9

    phinds

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    No, there is NOT. You are failing to take my advice that you have to think in terms of what exists in the 1D world. As I already said, failure to grasp this fundamental concept is quite common.
     
  11. Jul 25, 2016 #10
    Apologises for my ignorance and thank you for the help
     
  12. Jul 25, 2016 #11

    phinds

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    Ignorance is no vice if you correct it through the virtue of learning so you are doing well. I should have added that I had the exact same problem in understanding this when I was first introduced to it. In case you are not aware of it, I recommend the book Flatland.
     
  13. Jul 25, 2016 #12

    PeroK

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    The thing you are missing is the key word "minimum". I've underlined it above. You can use as many coordinates as you like to describe a sine wave or a line, but its dimension is the minimum needed. So, if there is any way to do it with one coordinate, it's one dimensional. In this case, see post #5, for example.

    Note that if you choose your x and y axes differently, then the sine wave could extend into 3D space and could be described using 3 coordinates. That doesn't make it a 3D object.
     
  14. Jul 25, 2016 #13

    chiro

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    Hey Einstein's Cat.

    If you assume that the normal ordered one-dimensional numbers are used to represent information then the dimension is the minimum number of those to represent it.

    Technically you could find a way to deform the space so that it's organized well enough to represent every state with one number (meaning you could take normal higher dimensional spaces and project them down to a single one without losing information about the space) but it's just the nature of mathematics to organize things spatially so that anything at right angles has its own component.

    To understand what I'm saying I'll restrict a three dimensional space to the integers from 0 to 100 inclusive on the x, y, and z axes. You will have 101^3 points for this space and there is no reason why you couldn't just have a one-dimensional number to represent every state.

    For the introduced example (in this post) you could use 101^2*a + 101*b + c where a,b,c are integers in the range 0 to 100. I have just transformed a higher dimensional space into a single number without losing information and you can do the same thing for other spaces provided that it is consistent to go between one and the other and that all states have been accounted for.
     
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