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Do Partial Spatial Dimensions Exist?

  1. Jul 20, 2011 #1
    So I'm sure everyone here knows of the basic spacial dimensions. 1D is a line, 2D a plane and 3D a cube. There is even a 4th dimension (theoretical), the tesseract. And an infinite number of dimensions beyond, represented by various hypercubes. Finding the space taken up by one of these objects (length, area, volume) is easy. The length of a line is x^1, the area of a plane is x^2, and the volume of a cube is x^3 continuing ad infinitum.

    But what about partial dimensions, do they exists (theoretically of course)? For example, a 2.5 dimension. It might make sense to think of dimensions as discrete, but then how would you account for the ability to find the space taken up by these objects (length, area, volume)? For example, you can find the "area/volume" of a 2.5 dimension "square/cube" by x^2.5. If the length of a side of this "square/cube" was 2 than 2^2.5 = 5.65685425 is the "area/volume" of this mythical partial dimensional object. If partial dimensions don't exists, what am I measuring?

    I have searched the internet far and wide and have found no mention of partial dimensions. Can someone please elaborate on them or if they are even possible? And if they are, could someone explain how I could go about drawing one?
     
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  3. Jul 20, 2011 #2

    DaveC426913

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    Read up on fractional dimensions, better known as fractals.

    The essence behind a fractional or fractal dimension is thus: on a straight line you have one degree of freedom - back and forth on the line. But kink that line up into a squiggly line, like a very craggy coastline, and you effectively have more than a single degree of freedom. The craggier the coastline gets, the closer you get to having a full two degrees of freedom. But you won't really reach two, it'll fall short, some fraction between 1 and 2 dimensions. This fraction is quantifiable and there is quite a bit of science developed around fractals.
     
  4. Jul 20, 2011 #3
    Wow that's complicated.

    So it seems that the dimension of fractals can be calculated, and they end up somewhere in between two dimensions. Can the reverse be done? Can I take a partial dimension and conceive a fractal? Can I conceive the fractals of any recognizable objects?
     
  5. Jul 21, 2011 #4
    I've been thinking about this some. Mathematically, time can be treated as a dimension. We also know that time is relative and approaches a stand still on the event horizon of a black hole. In this sense, as one approaches the event horizon of a black hole, four dimensional space-time would approach a three dimensional space. I would be interested if anyone here can comment on this.
     
  6. Jul 21, 2011 #5
    yes you are right time is not a geometrical dimension. It is only considered a dimension mathematically.

    I too think the same. But we should wait for an experts comment before reaching to conclusion.
     
  7. Jul 21, 2011 #6

    DaveC426913

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    Not true. It is very much a dimension. It just happens to be a timelike dimension, as opposed to a spacelike dimension.

    1] It only approaches a stand still from the point of view of a distant observer. For the observer at the horizon, time marches on. This is an issue of differing frames of reference, not of time "actually" stopping.

    2] To suggest that, because time appears to stop means that dimension disappears, is to likewise suggest that, if a ball's radius were kept uniform and unchanging along its x-axis (i.e. a cylinder), it has somehow become two dimensional.
     
  8. Jul 21, 2011 #7

    BWV

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    are the dimensions of Hilbert space infinitely countable or uncountable? i.e. is it an infinite number of discrete integer dimensions or a continuum across all the real numbers?
     
  9. Jul 21, 2011 #8
    I see your point on number 2. However, there is still the point of view of the distant observer. To the distant observer, the flow of time near an event horizon would be much less when compared to the local flow of time. If the local region of space-time had four dimensions, I could see an argument that the region of space-time near an event horizon has between three and four dimensions(comparatively speaking). I'm just brainstorming here. I'm not quite sure what a partial dimension would look like if it does indeed exist.
     
  10. Jul 21, 2011 #9
    You're getting very off topic. Can we please keep this to the spacial dimensions? Time is irrelevant here as I'm using dimensions as a mathematical concept, not a physical one.
     
  11. Jul 22, 2011 #10

    DaveC426913

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    Typically, once a question has been answered, a post is free to wander where its posters take it, at least until it gets locked for being way off.

    Has this not been answered? Do you have more questions?
     
  12. Jul 22, 2011 #11
    I don't quite get this concept. Whatever the shape of a line, it still only has one degree of freedom. Like, I could walk along the "squiggliest" coastline in the world and still parameterise it as the number of steps I've taken from my starting point.
     
  13. Jul 22, 2011 #12

    SpectraCat

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    Right, and the number of steps you take would also depend on the "squiggly-ness" of the coastline, so that is the second partial dimension. That is a simplification of the real mathematical answer, but I believe it captures the essence of the fractal dimension. Note also that such a fractal coastline must exist in a space of (at least) two dimensions. So, you can think of a fractal as a restricted exploration of the full two-dimensional space ... it's been a while since I read about this, but I believe that this is one way to calculate the dimension of a fractal. In other words, you calculate the fraction of the two dimensional space that can be accessed by moving along the fractal edge.
     
  14. Jul 22, 2011 #13
    I think I might have an idea what's going on. Essentially as the line becomes more squiggly it covers more two dimensional space while still only existing in one dimension. So in some limit of infinite-squigglyness, it should in theory cover every point in the plane?
     
  15. Jul 22, 2011 #14
    Typically, people use the Hausdorff dimension and not the number of degrees of freedom when talking about non-integer values of dimension. In this case, infinite squigglyness can have non-integer dimension.
     
  16. Jul 22, 2011 #15

    SpectraCat

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    Take a look at the book "Flatterland" (the sequel to the classic, "Flatland"), for an excellent and accessible introduction to fractal dimension, as well as many other mathematical topics related to topology and geometry.
     
  17. Jul 22, 2011 #16

    DaveC426913

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    Correct.

    Say I have one of those path-following robots (Black tape is placed on the white ground.The robot reads the tape with a sensor and is free to move along the line but nowhere else). It lives in a 20ft square room.

    The robot has one degree of freedom - back and forth along the line.

    But what if that line were very squiggly? The robot, despite the fact that it is constrained to one degree of freedom, can do pretty good job of getting most places in the room. (Practically speaking, the robot has a non-zero size, so can cover any spot in the room, however to be perfectly rigid, the robot has a centre point, which is zero size, that we are concerned with.)

    You can see that there is virtually no upper limit on how squiggly the line can be; the robot's freedom of movement approaches 2, though it never reaches it. To have exactly 2 dimensions of movement, the line of tape would have to be infinitely long.

    The squiggliness of the line can be associated with a value of its fractality, which will be a number somewhere between 1 and 2.
     
  18. Feb 25, 2012 #17
    So extrapolating from your example, can we assume we have full freedom of motion in 3 dimensions, and only partial freedom in the fourth? Does this suggest that in reality we live in the 3.5 dimension, or something of this nature due to relativistic effects that make parts of the fourth dimension inaccessible to us (as far as we know, at least)?
     
  19. Feb 26, 2012 #18
    Time can be considered a dimension but it really shouldn't be. It's kind of like calling tomatoes fruit. Especially since the fourth dimension has so many interesting properties, calling it time is a bit of a cop out. It's better just to call our perceivable universe spacetime and call the fourth dimension the fourth spatial dimension.

    You can think of time as a limited spacial dimension but it's going to reveal the exact reason why it doesn't make sense to call it a dimension.
     
  20. Feb 26, 2012 #19

    DaveC426913

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    Reveal for us.
     
  21. Feb 26, 2012 #20
    Just remember I'm not formally educated in any of this, I've just read books.

    Like I said before, the length of a 1D line is x^1, the area of a 2D plane is x^2, and the volume of a 3D cube is x^3. If we were to treat time as a whole dimension then the "hyper volume" of a 3D cube moving through time (4D space) would be determined by x^3. Which makes sense.

    But when you take into account that time only moves in one direction (or at least appears to for us) those equations don't really work. Can you imagine a volume moving in solely one direction? What would the "hyper volume" of that object be? To say that time is half a dimension gives a nonsensical answer. It's possible you can represent the single directionness of time some other way, but I'm pretty sure the volume should remain the same as a regular 4D object. As far as I can tell it really only makes sense to call time a whole dimension or none at all.
     
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