TITLE: Special properties of $\mathbb{R}^3$
QUESTION [7 upvotes]: Are there any special (nontrivial) properties of $\mathbb{R}^3$ that distinguish it from any other $\mathbb{R}^n$? If there are, what are some of the important ones?

REPLY [6 votes]: The unit sphere $\{x:\|x\|=1\}$ in $\mathbb R^3$ has the property that the area of each spherical slice 
$\{x:a\le x_1\le b\}$, $-1\le a\le b\le 1$, depends only on $b-a$. In more technical terms, the pushforward of the 
surface measure on a sphere under orthogonal projection to a line is a uniform measure on some segment. This property fails in all other dimensions. 
Nontrivial knots (informally speaking: smooth simple closed curves that cannot be continuously deformed to a circle) exist  in $\mathbb R^3$ but not in $\mathbb R^n$ for $n\ne 3$.