TITLE: When are $\mathbb{R}^n$ and $\mathbb{R}^m$ essentially similar?
QUESTION [9 upvotes]: Here is a rather vague and subjective question: for which $n$ and $m$ are 
$\mathbb{R}^n$ and $\mathbb{R}^m$ ``essentially similar''?  The answer depends partly on what type of
mathematician is answering it.
Of course, in a certain technical sense, $\mathbb{R}^n$ and $\mathbb{R}^m$ are different for any $n\neq m$ since they 
are not homeomorphic (or linearly isomorphic); thus they by definition differ in some describable topological (or algebraic) way.
But qualitatively all Euclidean spaces, especially ones of large dimension, seem to be ``similar.'' 
The question is really: how large does $n$ have to be for Euclidean spaces of dimension $\geq n$
all to behave essentially the same way?
I have included my own discussion of some possible answers in my Real Analysis manuscript at http://wolfweb.unr.edu/homepage/bruceb/Meas.pdf (Section XI.18.3).  I invite comments on this discussion.

REPLY [6 votes]: There are geometric and topological properties that significantly distinguish between Cartesian spaces of odd and even dimensions. Take, for example, the "sphere combing" property. The $(n-1)$-dimensional sphere (the boundary of a ball) in $\mathbb{R}^n$ admits a non-vanishing continuous tangent vector field if and only if $n$ is even.
Another striking difference is related to the existence of Hadamard matrices. It is known that an $n\times n$ Hadamard matrix exists only if $n=1,\ 2$, or is a multiple of $4$ (see https://en.wikipedia.org/wiki/Hadamard_matrix ), and Hadamard conjectured that there exists such a matrix for every $n$ divisible by $4$. The conjecture is still unsolved, but there exist many constructions producing an $n\times n$ Hadamard matrix for infinitely many values of $n$. Now, the geometric connection is this: Some $n+1$ vertices of the $n$-dimensional cube form a regular simplex if and only if there exists an $(n+1)\times(n+1)$ Hadamard matrix.
Thus, spaces $\mathbb{R}^n$ and $\mathbb{R}^{n+1}$ often differ essentially for infinitely many values of $n$, and I think many other examples of this nature can be readily presented.