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I have read that there exist non-integer fractal dimensions and the images generated from these dimensions look organic and they seem to provide a new way of describing the world around us, which raises the question. How many dimensions (spatially or otherwise) do we live in? 3, 3.1, 3.5 as well as 1 time dimension or could time itself exist in a fractal dimension of its own.

Qmechanic
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Jordan
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Off course the Idea of having fractal dimensions in our real physical word is extremely interesting, but till now, and despite of all the experiments made, there is no evidence of such a thing (for example, in LHC there is a dedicated team of physicists that analyzing colider data for any variation that may be treated as an evidence for extra dimensions).

Thus we can say for now, that within our current technologies and measurement apparatus accuracy, 4 desecrate dimensions are just enough to describe most of what we know about our universe

Anyway there is still many open questions that some physicists argue that they can be solved by considering fractal dimensions, but the most important part in those claims still missing: the experimental evidence, and we mast tend to search for the simplest explanations for those questions, and calming that it's a fractals is a very exotic one.

TMS
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The world is incredibly well-described locally by $\mathbb R^4$ and we know no experiment that would pretend the contrary.

  • That doesn't explain how we can generate fractals that are simply defined and yet look so much like the natural world. If parts of the real world can be found in fractals then couldn't it be possible that we exist in a fractal dimension. – Jordan Feb 04 '13 at 10:42
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    No, the physics is not described with fractals but with linear algebra and operators. And it works perfectly. –  Feb 04 '13 at 10:43
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    @JordanBrown An actual fractal has detail literally infinitely far down. In reality fractal structure will only go so far: eventually you get to molecules and that sort of thing. You don't need fractal dimensions to be able to have a repeating pattern! Fractal dimensions are just useful for talking about certain aspects of some repeating patterns. A fern leaf is very much 3D, even though the little sub-leaves look like copies of the big one. – Robert Mastragostino Feb 04 '13 at 10:56
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    This is actually not true. Quantum Physics hint that there are more, maybe infinite dimensions. Popular theories in String Physics hint 31 dimensions at a minimum. There is a theory that uses infinite dimensions to allow any possible universe. One could say we live in a 4-dimensional subspace of that. However certain results show there might be more than just that. – CBenni Feb 04 '13 at 10:58
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    For the moment the Standard Model is using only 4 dimensions. And this model is extremely accurate. –  Feb 04 '13 at 11:00
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    Define "extremely"? I would rather thing "sufficiently" is the correct word. –  Feb 04 '13 at 11:14
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    Extremely means : every time we are able to measure something more precisely, QFT gives the right answer. Every time. –  Feb 04 '13 at 11:16
  • I see. So you did mean "sufficiently". "Extremely" is a dangerous word, especially on a mathematics site. –  Feb 04 '13 at 11:22
  • Well I don't believe in coincidence. When I see such a model feeting so well the real world, I think it is not only 'a good approximation' –  Feb 04 '13 at 11:27
  • If you're going to be pedantic like that, $\mathbb{R}^{4}$ should be replaced with the appropriate local Riemannian manifold that locally gives the solar system and the galaxy and globally asymptotes to a flat Robertson-Walker spacetime. – Zo the Relativist Feb 04 '13 at 18:07