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What's special about 3 dimensions?

  1. May 8, 2010 #1
    I wonder why it took me so long to ask this question. But it's better late than never.

    Space has 3 dimensions. I have read numerous books where there are creatures who live in 1-D space, there're also hypothetical creatures who live in a 2-D space. But I don't remember reading a book where there are creatures who live in a 4-D space. In the numerous discussions I had with my friends, I remember (perhaps incorrectly) that we usually dropped the issue thinking that we being able to live in 3-D space can contemplate about 3-D spaces or lower; perhaps we can't contemplate about higher dimensioned spaces.

    But what is so fundamentally difficult about higher dimensioned spaces? Mathematically there's no problem. Just create a 4X1 vector to represent a point in 4-D space and so on.

    Maybe it's the inability to visualize that's making me queasy. But then there are some other things too. For instance, curl cannot be generalized to higher dimensioned space. Can you think of a computer game where 4-D spaces are used (visualization is not necessary, but the game should be truly 4-D in terms of SPACE)?

    Somehow I think I am on the wrong track or I am asking the wrong questions. Please let me know if you have a better insight.

    Regards,

    -sgsawant
     
  2. jcsd
  3. May 8, 2010 #2
    1d, 2d, 3d, 4d, etc... are just concepts. We can live in all of them or none of them depending on how you look at it. They are used to describe something and therefore they don't physically exist.

    I think 1D would be boring because you couldn't eat Fritos... but in 4d would they have more calories? I dunno, I will let a pro answer that.

    Rosie.
     
  4. May 8, 2010 #3
    Lift your hand up, and move it around two dimensions... then three.

    Now move it through a fourth dimension.

    You see the problem...
     
  5. May 8, 2010 #4

    Nabeshin

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    We as humans visualize things through experience and analogy. We "understand" the expansion of a universe when we equate it to an expanding balloon, something we have seen and are familiar with. It "makes sense" because we can create an analogy to something we can physically interact with. Similarly, we can physically interact with the two dimensional surface of a chalkboard, or a 1-dimensional line on the chalkboard, or a zero-dimensional point on the chalkboard. Who among us has ever interacted with a four-dimensional object? How could we possibly ever have a reference point to truly "understand" such a thing?

    That's what's fundamentally different: us.
     
  6. May 8, 2010 #5
    Yeah I agree. But then even the special theory of relativity is something that is different from what we usually see. Still I think I have understood it's space time implications.

    In fact the reason why I started this thread was that I find it fairly easy to imagine a game where special theory of relativity is implemented. For instance we can have a game like quake, cs or UT and then slow down light in that game. We can then allow players to have the option of running at speed near to that of light. What will happen in such a case? Nothing special. Just was usually happens. For players who have moved with velocities nearer to light, time will slow down as compared to the arena or map clock.

    Now my question is, is space the way we think of it, actually 3 dimensional? I mean not at the level where quantum mechanics becomes significant. There, to explain phenomenon we perhaps have to believe in higher dimensional co-ordinate systems. My question is about the space where day-to-day phenomena take place. Is there a 4th dimension of motion which we are not able to look at (look at as in not with just eyes - any sensor is ok) or is it truly 3 dimensional?
     
  7. May 9, 2010 #6
    Google the Poincare Conjecture, which was posited in the late 19th cent and recently proved at the beginning of this one.

    The main concequence for Physics of the Poincare Conjecture is that 3D is special, since it is mathematically special. If you understand curl and other vector mechanics you have enough mathematics to understand the Conjecture itslef.
     
  8. May 9, 2010 #7

    tom.stoer

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    There are some interesting things happening in mathematics of low dimensional manifolds.

    3-dim.space is the only space which allowes for the construction of a vector product; this is related to the existence of quaternions; in 7 dimension something similar is possible (due to the octonions) but compared to 3 dim. it's rather strange.

    Knots do only exist as embeddings of closed loops in 3-space. In 4-space (and higher) all knots are trivial.

    4-dim. space (or spacetime?) is the only case where the topological space Rn allowes for a continumm of inequivalent differential structures. In all other cases the topology of Rn defines one differential structure uniquely (that means homeomorphic manifolds are diffeomorphic automatically).

    Penrose argues that the existence of twistors is related to 4 dimensional spacetime - and vice versa.

    I think that only in 4 dimensions the symmetry group SO(N-1,1) of the spacetime manifold is not semi-simple, but a direct product SO(3,1)=SU(2)*SU(2).
     
  9. May 9, 2010 #8
    @ Studio & tom.stoer:

    Thanks a lot! You have exactly addressed my concern. I will try to read and understand Poincare's conjecture and try to understand its implications for 3D space (or space in general).

    @tom.stoer:

    No no... I was not hinting at space-time. I know how space-time is 4D, but what I am trying to understand is just the location of a point and why it's 3D in our world. You may argue that, even time is just a component of location, but that is not my point. My point is why there are just 3 other components other than time for location.

    Understanding Poincare's conjecture would be a big assignment in it itself, given my low level of expertise. :biggrin: I will be back after a few weeks.

    Regards,

    -sgsawant
     
  10. May 9, 2010 #9
    What's special is that the sphere is the only simply connected 3D manifold and (I think )only 3D has just the one.
     
  11. May 9, 2010 #10

    tom.stoer

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    Does that mean that in D>3 there are D-spheres plus other simply connected manifolds (w/o boundary) which are not D-spheres?
     
  12. May 10, 2010 #11
    Correct me if i'm wrong, but I don't think a line drawn using chalk is 1D. Cos no matter how thin you draw it, it will still have Length, breadth and height. Which probably means that unless the object is hypothetical, a 1D object is impossible. I can't imagine a 1D object, its impossible.
     
  13. May 10, 2010 #12
    The fact that curl cannot be generalised is a characteristic of the curl operator, not of the number of dimensions. It is defined for 3D vector fields and I can't see a significance to it not working outside its domain.
     
  14. May 10, 2010 #13
    Yes, it's hypothetical, as for something to actually exist, we'll have to have a real 'thickness', so if a 2D object somehow exists, its lingth is always 0, and so we can't pick it up, as, well, there's nothing to (But imagine how, if we were 4D creatures, we would visualise 3D objects to be impossible, as it is a 4D world, similarly as ours is a 3D one). Anyway, if your'e looking for a book, 'Hyperspace' by Michio Kaku covers interdimensional animals, concepts, theories etc.
     
  15. May 10, 2010 #14

    jack action

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    Never tried it, but I came across http://www.mushware.com/", once while web browsing. It seems to be a legit attempt to what you can expect of a 4-d game.

    There is also http://www.youtube.com/watch?v=6RxWVXfJBo8" that seems helpful to get used to play in 4-d.
     
    Last edited by a moderator: Apr 25, 2017
  16. May 10, 2010 #15
    Thanks :)
     
  17. May 10, 2010 #16

    tom.stoer

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    I am not talking about the curl = rot, but about the vector product

    [tex]vec{c} = \vec{a}\times\vec{b}[/tex]

    What one wants to do do is to construct a bilinear map from a vector space into the same vector space. This is possible only in three dimensions and it can be shown to be related to the existence of quaternions; quaterrnions are generalize compex numbers. They are a so-called division algebra, that means (roughly speaking) an algebra based on an invertable mutliplication. One can show that the only possible division algebras are R, C, Q, O; for the quaternions Q one has to drop the commutativity, for the octonions O the associativity. All other algebras are either isomorphic to one of these or do not allow for an inversion due to zero-divisors.

    It is suspected that the existence of octonions and the deep relation with the exceptional Lie algebra E8 (which has the largest "octonionic symmetry") plays some role for unified theories, e.g. string theory.

    In much the same way it could be possible that simpler structures like the quaternions single out 4 dim. spacetime. As of today there is no proof, but it's rather remarkable that 4 dim.and 8. dim spaces are rather unique in matehmatics, already at the "algebraic level".
     
  18. May 10, 2010 #17
    Tom, you will be telling us that the eigenvalues of random matrices match the Rieman Zeta function zeros next.
     
  19. May 10, 2010 #18

    tom.stoer

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    No, I'll publish the results next week in the New York Times :-)

    Honestly: there are some rather remarkable facts about mathematics in low dimensional spaces and there are some objects that do not have higher dimensional analogues. And up to now there are no convincing ideas why we live in three spatial dimensions (except for the anthropic principle which I think is not convincing per construction). So it is natural to look for reasons why three dimensions are singled out. I have now idea if these mathematical artefacts do have any physical implication, but it's interesting to think about it.
     
    Last edited: May 10, 2010
  20. May 10, 2010 #19
    I actually find it very hard to contemplate or believe there are true dimensions to space higher than 3. I think there is a real reason as to why the number 3, but I certain don't have that answer.
     
  21. May 10, 2010 #20

    tom.stoer

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    Nobody has this answer so far and it is unclear if it will ever be discovered, or - even worse - if it exists at all. Perhaps it's only anthropic reasoning that favours three dimensions, but I hope (!) that this is not the case. It would really be a paradigm shift in physics if arguments like "it is raining => the street is wet" are turned around as "the street is wet => perhaps it is raining".

    Therefore people try hard to find other reasons for this "why are there just three dimensions". I can't think about any other idea but a) mathematical or physical consistency or b) a dynamical principle e.g. in string theory that compactifies exactly 6 dimensions. But up to now there seems to be not even a rough idea about such a principle ...
     
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