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Premise: Any physically possible process of computations requires an underlying physical process. Such physical process can exist only in the available physical space, which can be modeled by the Euclidean $\mathbb{R}^3$ (most simplest model).

Question: What are some special properties of the Euclidean $\mathbb{R}^3$, which are very different from $\mathbb{R}^2$ and from $\mathbb{R}^4$ and from more dimensional metric spaces?

Example 1: I know that in the $\mathbb{R}^2$, an Euclidean minimum spanning tree (EMST) for a given set of points may be found in asymptotically optimal O(n log n) time using O(n) space. For more dimensions, finding optimal EMST algorithm remains an open problem. It is interesting to know, how $\mathbb{R}^2$ is different from $\mathbb{R}^3$.

Example 2: Another example is the max number of points interconnected (every-to-every) by lines of any curvature, such that no line intersect any other line. For $\mathbb{R}^2$ it is only 4 points (smth. like Mercedes symbol) - why 4 and not 3 or 5? How many points are possible to connect in such way in $\mathbb{R}^3$? (I suggest, infinite number, but it is interesting to look at a proof). What are some special properties of the Euclidean $\mathbb{R}^3$ such that the number of interconnected points jumps from 4 in $\mathbb{R}^2$ to infinity in $\mathbb{R}^3$?

Please excuse my non-professional questions, I am not mathematician.

Thank you in advance!

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I'm having difficulty understanding exactly what the question is here. The first example you give is (you say) an open problem and thus isn't a possible example. The second also you don't know but doesn't seem related to the title of the question. –  Andrew Stacey Dec 18 '09 at 15:59
    
@Andrew: Second example is related to the question "What are some special properties of the Euclidean R^3 such that the number of interconnected points jumps from 4 in R^2 to infinity in R^3? –  psihodelia Dec 18 '09 at 16:09
    
I think the title could lead to an interesting question (probably several interesting questions), but this isn't it. So I'm not voting to close, but if someone who (unlike me) actually knew some geometry edited the question, I could be very interested in it. –  Harrison Brown Dec 18 '09 at 16:24
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Even with the edit, it's still far too vague for my liking. Any maths that is there could go in the dimension leaps question. There's always a progression as one moves up dimension and often the trick is to find the correct generalisation: lines in each dimension just isn't it. Looking for qualitative leaps was the point of my "dimension leaps" question, but this seems more focussed on quantitative differences, which aren't (to me) as interesting. @Harrison: I don't disagree, but in the meantime I'm voting to close as I don't think that this question is it. –  Andrew Stacey Dec 18 '09 at 16:34
    
@Andrew, @Harrison: Maybe should I divide this too vague post into 2? Example 2 is very interesting to be discussed. –  psihodelia Dec 18 '09 at 17:15
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closed as not a real question by Andrew Stacey, Reid Barton, David Speyer, Ben Webster Dec 18 '09 at 17:28

It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.