**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!

dimension leapsquestion. 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 thatthisquestion is it. – Andrew Stacey Dec 18 '09 at 16:34