High-dimensionality and intuition 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)? Thanks! 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.

Intuitions fails in higher-dimensions:

* Imagine a unit hyper-sphere within a cube with side 2. In low dimensions (2d), most of the volume (area) is within the hyper-sphere (circle) and only a small fraction of the volume is outside of the hyper-sphere, thus in the corners of the hyper-cube (square). However, for high dimensions it is the other way around. The volume of the hyper-cube is obviously $V_q = 2^n$ while the volume of the unit hyper-sphere is $V_s=\frac{\pi^{\frac{n}{2}}}{(\frac{n}{2})!}$ (for even $n$) with $\lim_{n\rightarrow \infty} \frac{\pi^{\frac{n}{2}}}{(\frac{n}{2})!}=0$. In other words: Only for low dimensions, the bounding box of a hyper-sphere is a 'fair' approximation of the volume of the sphere.

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