Over the years I've come across (usually as a tangential remark in a lecture) examples of how our intuitions (derived as they are from the experience of living in 3-dimensional space) will lead us badly astray when thinking about some $n$-dimensional Euclidean space, for some $n > 3$, especially if $n \gg 3$.
Does anyone know of a compendium of these "false intuitions" (in high-dimensional Euclidean space)?
P.S. The motivation for this question is more than amusement. In my line of work, the geometrization of a problem by mapping it onto some Euclidean $n$-space is often seen as a boon to intuition, even when $n$ is huge. I suspect, however, that the net gain in intuition resulting from this maneuver may very well be negative! In any case, it seems like a good idea to be conversant with those intuitions that should be disregarded.