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Each dimension is perpendicular at all another

  1. Jun 22, 2004 #1
    Does this concept is connected with a pure geometry (of space)?.
    Thus each dimension is perpendicular at all another.
    It is simple for presenting graphically up to three dimensions:
    point, line, square, cube and …..
    Further it is much more difficult. What a physical sense of various dimensions?
     
  2. jcsd
  3. Jun 22, 2004 #2
    First of all, each dimension doesn't have to be perpendicular. What defines a dimension is that the basis vectors can't be a linear combination of each other.

    If you are really interrested try reading some linear algebra. that will give you a good idea of what is a space, subspace, dimension, etc.
     
  4. Jun 23, 2004 #3
    But for the three dimension it works pretty good.
    What is the reason for change of a principle of perpendicularity for higher dimensions?
    It seems not logical at least.
     
  5. Jun 23, 2004 #4

    HallsofIvy

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    I don't understand what you mean. There is no "change of a principle of perpendicularity for higher dimensions". The general, linear algebra definition, of 'dimension n' is that there exist n independent vectors. As long as you are working in an inner product space in which "perpendicularity" can be defined, it is always possible to choose a basis of n perpendicular vectors, corresponding to n perpendicular directions.
     
  6. Jun 23, 2004 #5
    Then what , according to linear algebra, the fourth dimension look like?
    Is it not the sphere? The adding of next dimension add a corresponding move at object
    It seems the geometry of space is a dynamic geometry.
     
  7. Jun 23, 2004 #6
    Maybe you're not getting it. A 2-d space can be made up of x^2 + x^3.
    A dimension is a purely mathematical concept.... Even in physics.

    Now to really get your noodle. You can transform a set of vectors to another set of vectors. A NullSpace is the set of all vectors that get mapped to 0 vector.
     
  8. Jun 23, 2004 #7

    Moe

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    Michael, the problem is that you can't picture more than three dimensions. I have seen representations of hypercubes (4-D cubes), but they look all weird. (For all you Rubic's cube fans out there, I found a little flash applet with a 4-D rubic's cube.) You just have to accept them as a mathematical concept. In three-D space, a set of base vectors would be
    1 0 0
    0 1 0
    0 0 1

    These can be graphically represented as the x, y and z axis.
    A base system for a 4-D space would be

    1 0 0 0
    0 1 0 0
    0 0 1 0
    0 0 0 1

    They are all "perpendicular", multiplying any two of them will give you 0. But finding a graphic representation of those is pretty much impossible.
     
  9. Jun 23, 2004 #8
    But I do not see nor a zero point nor a mirror reflection of each dimension in your matrixes.
    A zero point, (x,-x), (y,-y), (z,-z) are always present on any real graph.
    Here I see your problems, not mine.
     
  10. Jun 23, 2004 #9

    Integral

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    Michale,

    A reflection is an operation performed on an object, it is not something that is seen in the basis vectors themselves.
     
  11. Jun 24, 2004 #10
    Let's check up.
    The LINE this a propagation of POINT. This is an operation.
    The SQUARE this a propagation of a line. This is an operation.
    A CUBE this a propagation of the square. This is an operation.
    The SPHERE this recurrence of a cube at his rotation concerning all three coordinates. Thus the sphere itself rotates also, but in another mode. This is obviously an operation.
    We can observe this operations in the nature.
     
  12. Jun 24, 2004 #11

    Moe

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    What do you mean, you don't see a zero point reflection in my matrices? It works just like the three-D thing. You have to stop trying to visualize something with 4 dimensions.
    This
    [​IMG] is a hypercube, a 4-D cube. The reason why it looks so weird is that it is essentially impossible to visualize it. Our mind works in only three dimensions.


    Edit: How come image [​IMG]
     
  13. Jun 24, 2004 #12
    yea i find it pretty hard to wrap my head around the whole issue of "11 dimensions" and the likes, but H.G. Wells described dimensions vivedly in the prologue to 'Time Machine' the original novel, give it crack cos' its hood rich
     
  14. Jun 24, 2004 #13
     
  15. Jun 24, 2004 #14
    I always thought that we exist in the TIME too. Forgive my provincial naivety, please.
     
  16. Jun 24, 2004 #15

    Moe

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    Now you have lost me.

    We exist in all dimensions. That doesn't mean we can understand them. And in mathematics, you can have as many dimensions as you like. There is no need for a physical representation of those dimensions. For example, it might be feasible to express certain characteristics as a multi-dimensional vector. In Relativity, you use 4-D vectors, one component is time, the other three are speeds, or one is energy, the other three are linear momenta. Stop trying to visualize it - it won't work.

    Now, if you want to, you can view time as the fourth dimension. In that case you could maybe visualize a 4-D cube by looking at a video of a 3-D cube that is being deformed. But what would a 5-D cube look like then? So let me rephrase my original sentence:
    Our mind works in three spatial dimensions only.
     
  17. Jun 24, 2004 #16
    Why, if it turns out the pretty successful?
     
  18. Jun 24, 2004 #17
    Then 5-D (and all higher) an OBJECT would look like the sphere too.
    I do not remember who had said first about “ higher spheres” , but this the words are prophetical
     
  19. Jun 24, 2004 #18

    Moe

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    Why would an object with more than three dimensions look like a sphere?
     
  20. Jun 25, 2004 #19
    Moe,
    I have explained this for 4-D object in a post #10 of thread.
    For the following dimensions :
    Any rotation of sphere gives the sphere in a result.
    Hence any an object with more than four dimensions look like a sphere also.
    They are invisible for us, but we can perceive their some manifestation.
     
  21. Jun 28, 2004 #20
    It is possible to consider dimensions as evolution of the universe from nothing.
    Zero point--> ray of light--> plane of light--> volume of light -->sphere of light--> rotation of spheres of light.
    But all aspires to return to a zero point. It corresponds to gravitation.
     
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